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A small square loop of side ‘\( a \)’ and one turn is placed inside a larger square loop of side \( b \) and one turn \( (b \gg a) \). The two loops are coplanar with their centres coinciding. If a current \( I \) is passed in the square loop of side ‘\( b \)’, then the coefficient of mutual inductance between the two loops is:

Given below are two statements:
Statement I: The electric force changes the speed of the charged particle and hence changes its kinetic energy; whereas the magnetic force does not change the kinetic energy of the charged particle.
Statement II: The electric force accelerates the positively charged particle perpendicular to the direction of electric field. The magnetic force accelerates the moving charged particle along the direction of magnetic field.
In the light of the above statements, choose the most appropriate answer from the options given below:

An electron with energy \( 0.1 \,\text{keV} \) moves at right angle to the earth’s magnetic field of \( 1\times 10^{-4}\, \text{Wb m}^{-2} \). The frequency of revolution of the electron will be (Take mass of electron \( =9.0\times 10^{-31}\,\text{kg} \))

There are two infinitely long straight current carrying conductors and they are held at right angles to each other so that their common ends meet at the origin as shown in the figure given below. The ratio of current in both conductors is \( 1:1 \). The magnetic field at point \( P \) is:

A proton, a deuteron and an \( \alpha \)- particle with same kinetic energy enter into a uniform magnetic field at right angle to magnetic field. The ratio of the radii of their respective circular paths is:

In a co-axial straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero:

Match List I with List II.

Choose the correct answer from the options given below:

With the help potentiometer, we can determine the value of emf of given cell. The sensitivity of the potentiometer is:
(A) Directly proportional to the length of the potentiometer wire.
(B) Directly proportional to the potential gradient of the wire.
(C) Inversely proportional to the potential gradient of the wire.
(D) Inversely proportional to the length of the potentiometer wire.
Choose the correct option for the above statements

A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of \( 2 \times 10^{5} \, \text{m s}^{-1} \). When the electric field is switched off, the proton moves along a circular path of radius \( 2 \, \text{cm} \). The magnitude of electric field is \( x \times 10^{4} \, \text{N/C} \). The value of \( x \) is __________. Take the mass of the proton \( = 1.6 \times 10^{-27} \, \text{kg} \).

A current of \(200\,\mu\) \(\text{A}\) deflects the coil of a moving coil galvanometer through \(60^{\circ}\). The current to cause deflection through \(\displaystyle\frac{\pi}{10}\) radian is:

A charge particle of \( 2\mu \text{C} \) accelerated by a potential difference of \( 100 \, \text{V} \) enters a region of uniform magnetic field of magnitude \( 4 \, \text{mT} \) at right angle to the direction of field. The charge particle completes semicircle of radius \( 3 \, \text{cm} \) inside magnetic field. The mass of the charge particle is \( \_\_\_\_\_ \times 10^{-18} \) \( \text{kg} \).

A current of \( 5 \, \text{A} \) exists in a square loop of side \( \frac{1}{\sqrt{2}} \, \text{m} \). Then the magnitude of the magnetic field \( B \) at the centre of the square loop will be \( p \times 10^{-6} \, \text{T} \), where, value of \( p \) is ________.
[ Take \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m A}^{-1} \) ].

\( 1 \, \mu\text{C} \) charge moving with velocity \( \vec{v}=(\hat{i}-2\hat{j}+3\hat{k})\, \text{m/s} \) in the region of magnetic field \( \vec{B}=(2\hat{i}+3\hat{j}-5\hat{k})\, \text{T} \). The magnitude of force acting on it is \( \sqrt{\alpha}\times 10^{-6} \, \text{N} \). The value of \( \alpha \) is ________.

A deuteron and a proton moving with equal kinetic energy enter into to a uniform magnetic field at right angle to the field. If \( r_{d} \) and \( r_{p} \) are the radii of their circular paths respectively, then the ratio \( \frac{r_{d}}{r_{p}} \) will be \( \sqrt{x} : 1 \) where \( x \) is ________.

Consider a long thin conducting wire carrying a uniform current \( I \). A particle having mass \( ''M'' \) and charge \( ''q ''\) is released at a distance \( ''a'' \) from the wire with a speed \( v_0 \) along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance \( x \) from the wire. The value of \( x \) is: [\( \mu_0 \) is vacuum permeability]

Two short magnetic dipoles \( m_1 \) and \( m_2 \) each having magnetic moment of \( 1 \, \text{A m}^2 \) are placed at point \( O \) and \( P \) respectively. The distance between \( OP \) is \( 1 \, \text{meter} \). The torque experienced by the magnetic dipole \( m_2 \) due to the presence of \( m_1 \) is ________ \( \times 10^{-7} \) \( \text{N m} \).

Water boils in an electric kettle in \( 20 \) minutes after being switched on. Using the same main supply, the length of the heating element should be __________ to __________ times of its initial length if the water is to be boiled in \( 15 \) minutes.

Two ions having same mass have charges in the ratio \( 1:2 \). They are projected normally in a uniform magnetic field with their speeds in the ratio \( 2:3 \). The ratio of the radii of their circular trajectories is:

Two long parallel wires \( X \) and \( Y \), separated by a distance of \( 6 \, \text{cm} \), carry currents of \( 5 \, \text{A} \) and \( 4 \, \text{A} \), respectively, in opposite directions as shown in the figure. Magnitude of the resultant magnetic field at point \( P \) at a distance of \( 4 \, \text{cm} \) from wire \( Y \) is \( x \times 10^{-5} \, \text{T} \). The value of \( x \) is __________. Take permeability of free space as \( \mu_0 = 4\pi \times 10^{-7} \) SI units.

A closely wounded circular coil of radius \( 5 \, \text{cm} \) produces a magnetic field of \( 37.68 \times 10^{-4} \, \text{T} \) at its center. The current through the coil is __________ \( \text{A} \).
[Given, number of turns in the coil is \( 100 \) and \( \pi = 3.14 \) ]

A proton and an \( \alpha -\)particle, having kinetic energies \( K_{p} \) and \( K_{\alpha} \) respectively, enter into a magnetic field at right angles. The ratio of the radii of trajectory of proton to that of \( \alpha \)–particle is \( 2:1 \). The ratio of \( K_{p} : K_{\alpha} \) is:

The free space inside a current carrying toroid is filled with a material of susceptibility \( 2 \times 10^{-2} \). The percentage increase in the value of magnetic field inside the toroid will be:

A wire \( X \) of length \( 50\,\text{cm} \) carrying a current of \( 2\,\text{A} \) is placed parallel to a long wire \( Y \) of length \( 5\,\text{m} \). The wire \( Y \) carries a current of \( 3\,\text{A} \). The distance between two wires is \( 5\,\text{cm} \) and currents flow in the same direction. The force acting on the wire \( Y \) is

If the maximum value of accelerating potential provided by a radio frequency oscillator is \( 12 \, \text{kV} \). The number of revolution made by a proton in a cyclotron to achieve one sixth of the speed of light is _____.
[ \( m_p = 1.67 \times 10^{-27} \, \text{kg} \), \( e = 1.6 \times 10^{-19} \, \text{C} \), Speed of light \( = 3 \times 10^{8} \, \text{m/s} \) ] .

In a uniform magnetic field, the magnetic needle has a magnetic moment \( 9.85\times 10^{-2} \, \text{A} / \text{m}^{2} \) and moment of inertia \( 5\times 10^{-6} \, \text{kg m}^{2} \). If it performs \( 10 \) complete oscillations in \( 5 \) seconds then the magnitude of the magnetic field is ________ \( \text{mT} \). [Take \( \pi^{2} \) as \( 9.85 \)]

A beam of light travelling along \( X \)-axis is described by the electric field \( E_{y} = 900 \sin \omega (t - x/c) \). The ratio of electric force to magnetic force on a charge \( q \) moving along \( Y \)-axis with a speed of \( 3 \times 10^{7} \ \text{m s}^{-1} \) will be:
(Given speed of light \( = 3 \times 10^{8} \ \text{m s}^{-1} \))

A deuteron and an alpha particle having equal kinetic energy enter perpendicularly into a magnetic field. Let \( r_d \) and \( r_\alpha \) be their respective radii of circular path. The value of \( \dfrac{r_d}{r_\alpha} \) is

Figure shows a current carrying square loop ABCD of edge length is 'a' lying in a plane. If the resistance of the ABC part is \( r \) and that of ADC part is \( 2r \), then the magnitude of the resultant magnetic field at centre of the square loop is:

The current passing through a conducting loop in the form of equilateral triangle of side \( 4\sqrt{3}\ \text{cm} \) is \( 2\,\text{A} \). The magnetic field at its centroid is \( a \times 10^{-5}\ \text{T} \). The value of \( a \) is ______.
(Given: \( \mu_{0} = 4\pi \times 10^{-7} \) SI units)

A small cube of side \( 1 \, \text{mm} \) is placed at the centre of a circular loop of radius \( 10 \, \text{cm} \) carrying a current of \( 2 \, \text{A} \). The magnetic energy stored inside the cube is \( \alpha \times 10^{-14} \, \text{J} \). The value of \( \alpha \) is _____.
\( (\mu_{0}=4\pi \times 10^{-7} \, \text{T m/A} , \ \pi = 3.14) \)

A square loop of edge length \( 2 \, \text{m} \) carrying current of \( 2 \, \text{A} \) is placed with its edges parallel to the \( x\text{-}y \) axis. A magnetic field is passing through the \( x\text{-}y \) plane and expressed \( \vec{B} = B_{0}(1 + 4x) \, \hat{k} \), where \( B_{0} = 5 \, \text{T} \). The net magnetic force experienced by the loop is _______ \( \text{N} \).

A hairpin like shape as shown in figure is made by bending a long current carrying wire. What is the magnitude of a magnetic field at point \( P \) which lies on the centre of the semicircle?

A charge particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charge particle increases to \( 4 \) times its initial value. What will be the ratio of new radius to the original radius of circular path of the charge particle:

A single turn current loop in the shape of a right angle triangle with sides \( 5 \, \text{cm} \), \( 12 \, \text{cm} \), \( 13 \, \text{cm} \) is carrying a current of \( 2 \, \text{A} \). The loop is in a uniform magnetic field of magnitude \( 0.75 \, \text{T} \) whose direction is parallel to the current in the \( 13 \, \text{cm} \) side of the loop. The magnitude of the magnetic force of the \( 5 \, \text{cm} \) side will be \( \frac{x}{130} \, \text{N} \). The value of \( x \) is ________.

The magnetic field inside a \(200\) turns solenoid of radius \(10\, \text{cm}\) is \(2.9 \times 10^{-4}\, \text{Tesla}\). If the solenoid carries a current of \(0.29\, \text{A}\), then the length of the solenoid is ________ \( \pi \) \( \text{cm} \).

A coil having \( N \) turns is wound tightly in the form of a spiral with inner and outer radii ‘\( a \)’ and ‘\( b \)’ respectively. Find the magnetic field at centre, when a current \( I \) passes through coil:

A circular loop of radius \( r \) is carrying current \( 1 \, \text{A} \). The ratio of magnetic field at the centre of circular loop and at a distance \( r \) from the center of the loop on its axis is:

The current sensitivity of a galvanometer can be increased by:
(a) decreasing the number of turns
(b) increasing the magnetic field
(c) decreasing the area of the coil
(d) decreasing the torsional constant of the spring
Choose the most appropriate answer form the options given below:

Two long current carrying conductors are placed parallel to each other at a distance of \( 8 \ \text{cm} \) between them. The magnitude of magnetic field produced at mid-point between the two conductors due to current flowing in them is \( 300 \ \mu \text{T} \). The equal current flowing in the two conductors is:

A long straight wire with a circular cross-section having radius \( R \), is carrying a steady current \( I \). The current \( I \) is uniformly distributed across this cross-section. Then the variation of magnetic field due to current \( I \) with distance \( r \) (\( r<R \)) from its centre will be:

A heater is designed to operate with a power of \( 1000 \,\text{W} \) in a \( 100 \,\text{V} \) line. It is connected in combination with a resistance of \( 10\,\Omega \) and a resistance \( R \), to \( 100 \,\text{V} \) mains as shown in figure. For the heater to operate at \( 62.5 \,\text{W} \), the value of \( R \) should be ______ \( \Omega \).

Figure A and B show two long straight wires of circular cross-section (\( a \) and \( b \) with \( a < b \)), carrying current \( I \) which is uniformly distributed across the cross-section. The magnitude of magnetic field \( B \) varies with radius \( r \) and can be represented as:

The magnetic field at the centre of a wire loop formed by two semicircular wires of radii \( R_1 = 2\pi \, \text{m} \) and \( R_2 = 4\pi \, \text{m} \), carrying current \( I = 4 \, \text{A} \) as per figure given below is \( \alpha \times 10^{-7} \, \text{T} \). The value of \( \alpha \) is __________ . (Centre O is common for all segments).

A long solenoid carrying a current produces a magnetic field \( B \) along its axis. If the current is doubled and the number of turns per cm is halved, the new value of magnetic field will be equal to:

Two long, straight wires carry equal currents in opposite directions as shown in figure. The separation between the wires is $5.0\text{ cm}$. The magnitude of the magnetic field at a point P midway between the wires is $\underline{\quad}\ \mu\text{T}$ . (Given : $\mu_0 = 4\pi \times 10^{-7}\text{ T m A}^{-1}$)

A rigid wire consists of a semicircular position of radius \( R \) and two straight sections. The wire is partially immerged in a perpendicular magnetic field \( B = B_0 \hat{k} \) as shown in figure. The magnetic force on the wire if it has current \( i \) is:

An electron moves through a uniform magnetic field $\vec{B} = B_0\hat{i} + 2B_0\hat{j}\text{ T}$. At a particular instant of time, the velocity of electron is $\vec{u} = 3\hat{i} + 5\hat{j}\text{ m/s}$. If the magnetic force acting on electron is $\vec{F} = 5e\hat{k}\text{ N}$, where $e$ is the charge of electron, then the value of $B_0$ is ________ $\text{T}$.

Statement I: The ferromagnetic property depends on temperature. At high temperature ferromagnet becomes paramagnet.
Statement II: At high temperature, the domain wall area of a ferromagnetic substance increases.
In the light of the above statements, choose the most appropriate answer from the options given below

Two long parallel conductors \( S_1 \) and \( S_2 \) are separated by a distance \( 10 \, \text{cm} \) and carrying currents of \( 4 \, \text{A} \) and \( 2 \, \text{A} \) respectively. The conductors are placed along x-axis in X-Y plane. There is a point \( P \) located between the conductors (as shown in figure). A charge particle of \( 3\pi \) coulomb is passing through the point \( P \) with velocity \( \vec{v} = (2 \hat{i} + 3 \hat{j}) \, \text{m/s} \); where \( \hat{i} \) & \( \hat{j} \) represents unit vector along \( x \) & \( y \) axis respectively. The force acting on the charge particle is \( 4 \pi \times 10^{-5} \, (-x \hat{i} + 2 \hat{j}) \, \text{N} \). The value of \( x \) is:

A single current carrying loop of wire carrying current \( I \) flowing in anticlockwise direction seen from \( +ve \) \( z \) direction and lying in \( xy \) plane is shown in figure. The plot of \( \, \hat{\jmath} \, \) component of magnetic field \( (B_y) \) at a distance \( a\) (less than radius of the coil) and on \( yz \) plane vs \( z \) coordinate looks like:

Two circular coils P and Q of \( 100 \) turns each have same radius of \( \pi \, \text{cm} \). The currents in P and R and \( 1 \, \text{A} \) and \( 2 \, \text{A} \) respectively. P and Q are placed with their planes mutually perpendicular with their centres coincide. The resultant magnetic field induction at the centre of the coils is \( \sqrt{x} \, \text{mT} \), where \( x \) _______.
[Use \( \mu_{0} = 4 \pi \times 10^{-7} \, \text{T m A}^{-1} \)]

A loop of flexible wire of irregular shape carrying current is placed in an external magnetic field. Identify the effect of the field on the wire.

A \( 2 \, \text{A} \) current carrying straight metal wire of resistance \( 1 \, \Omega \), resistivity \( 2 \times 10^{-6} \, \Omega \text{m} \), area of cross-section \( 10 \, \text{mm}^2 \) and mass \( 500 \, \text{g} \) is suspended horizontally in mid air by applying a uniform magnetic field \( \vec{B} \). The magnitude of \( B \) is __________ \( \times 10^{-1} \, \text{T} \) (given, \( g = 10 \, \text{m/s}^2 \)).

The elastic potential energy stored in a steel wire of length \( 20 \, \text{m} \) stretched through \( 2 \, \text{cm} \) is \( 80 \, \text{J} \). The cross sectional area of the wire is ________ \( \text{mm}^2 \). (given \( y = 2.0 \times 10^{11} \, \text{Nm}^{-2} \))

A circular coil of radius \( 2 \, \text{cm} \) and \( 125 \) turns carries a current of \( 1 \, \text{A} \). The coil is placed in a uniform magnetic field of magnitude \( 0.4 \, \text{T} \). The axis of the coil makes an angle of \( 30^\circ \) with the direction of the magnetic field. The torque acting on the coil is \( \alpha \times 10^{-4} \, \text{N m} \). The value of \( \alpha \) is ______. \( (\pi = 3.14) \)

A moving coil galvanometer has \( 100 \) turns and each has an area of \( 2.0 \, \text{cm}^2 \). The magnetic field produced by the magnet is \( 0.01 \, \text{T} \) and the deflection in the coil is \( 0.05 \) radian when a current of \( 10 \, \text{mA} \) is passed through it. The torsional constant of the suspension wire is \( x \times 10^{-5} \, \text{N·m/rad} \). The value of \( x \) is __________.

Two long straight wires \( P \) and \( Q \) carrying equal current \( 10 \, \text{A} \) each were kept parallel to each other at \( 5 \, \text{cm} \) distance. Magnitude of magnetic force experienced by \( 10 \, \text{cm} \) length of wire \( P \) is \( F_1 \). If distance between wires is halved and currents on them are doubled, force \( F_2 \) on \( 10 \, \text{cm} \) length of wire \( P \) will be:

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: If Oxygen ion \( \mathrm{O}^{-2} \) and Hydrogen ion \( (\mathrm{H}^{+} )\) enter normal to the magnetic field with equal momentum, then the path of \( \mathrm{O}^{-2} \) ion has a smaller curvature than that of \( \mathrm{H}^{+} \).
Reason R: A proton with same linear momentum as an electron will form a path of smaller radius of curvature on entering a uniform magnetic field perpendicularly.
In the light of the above statements, choose the correct answer from the options given below:

A current of \(\ 1.5 \, \text{A} \) is flowing through a triangle, of side \(\ 9 \, \text{cm} \) each. The magnetic field at the centroid of the triangle is: (Assume that the current is flowing in the clockwise direction.)

Two ions of masses \( 4 \, \text{amu} \) and \( 16 \, \text{amu} \) have charges \( +2e \) and \( +3e \) respectively. These ions pass through the region of constant perpendicular magnetic field. The kinetic energy of both ions is same. Then:

A proton with a kinetic energy of \( 2.0 \, \text{eV} \) moves into a region of uniform magnetic field of magnitude \( \dfrac{\pi}{2} \times 10^{-3} \, \text{T} \). The angle between the direction of magnetic field and velocity of proton is \( 60^\circ \). The pitch of the helical path taken by the proton is ________ \( \text{cm} \). (Take, mass of proton \( = 1.6 \times 10^{-27} \, \text{kg} \) and Charge on proton \( = 1.6 \times 10^{-19} \, \text{C} \).)

An aluminium rod with Young’s modulus \( Y = 7.0 \times 10^{10} \, \text{N}/\text{m}^{2} \) undergoes elastic strain of \( 0.04\% \). The energy per unit volume stored in the rod in SI unit is:

A regular polygon of \( 6 \) sides is formed by bending a wire of length \( 4\pi \) meter. If an electric current of \( 4\pi \sqrt{3} \, \text{A} \) is flowing through the sides of the polygon, the magnetic field at the centre of the polygon would be \( x \times 10^{-7} \, \text{T} \). The value of \( x \) is ______.

The magnetic field at the centre of a circular coil of radius \( r \), due to current \( I \) flowing through it, is \( B \). The magnetic field at a point along the axis at a distance \( \frac{r}{2} \) from the centre

The length of a metallic wire is increased by \( 20\% \) and its area of cross section is reduced by \( 4\% \). The percentage change in resistance of the metallic wire is ______________.

The magnitude of magnetic induction at mid point \( O \) due to current arrangement as shown in figure will be:

The fractional change in the magnetic field intensity at a distance \( r \) from centre on the axis of current carrying coil of radius\( a \) to the magnetic field intensity at the centre of the same coil is: (Take \( r \ll a \)).

A uniform magnetic field of \( 2 \times 10^{-3} \, \text{T} \) acts along positive \( Y \)-direction. A rectangular loop of sides \( 20 \, \text{cm} \) and \( 10 \, \text{cm} \) with current of \( 5 \, \text{A} \) is in \( Y - z \) plane. The current is in anticlockwise sense with reference to negative \( X \) axis. Magnitude and direction of the torque is:

The horizontal component of earth's magnetic field at a place is \( 3.5 \times 10^{-5} \, \text{T} \). A very long straight conductor carrying current of \( \sqrt{2} \, \text{A} \) in the direction from South-East to North-West is placed. The force per unit length experienced by the conductor is ______ \( \times 10^{-6} \, \text{N} / \text{m} \).

A null point is found at \( 200 \, \text{cm} \) is potentiometer when cell in secondary circuit is shunted by \( 5\,\Omega \). When a resistance of \( 15\,\Omega \) is used for shunting, null point moves to \( 300 \, \text{cm} \). The internal resistance of the cell is __________ \( \Omega \).

As shown in the figure, a metallic rod of linear density \( 0.45 \, \text{kg} \, \text{m}^{-1} \) is lying horizontally on a smooth inclined plane which makes an angle of \( 45^\circ \) with the horizontal. The minimum current flowing in the rod required to keep it stationary, when \( 0.15 \, \text{T} \) magnetic field is acting on it in the vertical upward direction, will be:
{Use \( g = 10 \, \text{m/s}^2 \)}

\( B_X \) and \( B_Y \) are the magnetic field at the centre of two coils \( X \) and \( Y \) respectively each carrying equal current. If coil \( X \) has \( 200 \) turns and \( 20 \, \text{cm} \) radius and coil \( Y \) has \( 400 \) turns and \( 20 \, \text{cm} \) radius, the ratio of \( B_X \) and \( B_Y \) is:

A straight wire carrying a current of \( 14 \, \text{A} \) is bent into a semicircular arc of radius \( 2.2 \, \text{cm} \) as shown in the figure. The magnetic field produced by the current at the centre \( (O) \) of the arc, is ______ \( \times 10^{-4} \, \text{T} \).

A galvanometer has a coil of resistance \( 200 \, \Omega \) with a full scale deflection at \( 20 \, \mu \text{A} \). The value of resistance to be added to use it as an ammeter of range \( (0 - 20) \, \text{mA} \) is:

The electric current in a circular coil of \( 2 \) turns produces a magnetic induction \( B_1 \) at its centre. The coil is unwound and is rewound into a circular coil of \( 5 \) turns and the same current produces a magnetic induction \( B_2 \) at its centre. The ratio of \( \dfrac{B_2}{B_1} \) is:

As shown in the figure, a long straight conductor with semicircular arc of radius \( \frac{\pi}{10} \, \text{m} \) is carrying current \( I = 3 \, \text{A} \). The magnitude of the magnetic field, at the centre O of the arc is:
(The permeability of the vacuum \( = 4\pi \times 10^{-7} \, \text{N A}^{-2} \).)

A charge particle moving in magnetic field \( B \), has the components of velocity along \( B \) as well as perpendicular to \( B \). The path of the charge particle will be:

Which of the following statements are correct?
\( (A) \) Electric monopoles do not exist whereas magnetic monopoles exist.
\( (B) \) Magnetic field lines due to solenoid at its ends and outside cannot be completely straight and confined.
\( (C) \) Magnetic field lines are completely confined within a toroid.
\( (D) \) Magnetic field lines inside a bar magnet are not parallel.
\( (E) \) \( \chi = -1 \) is the condition for a perfect diamagnetic material, where \( \chi \) is its magnetic susceptibility.
Choose the correct answer from the options given below.

Find the magnetic field at the point P in figure. The curved portion is a semicircle connected to two long straight wires.

Choose the correct option:

A charge \( Q \) is moving \( \vec{dl} \) distance in the magnetic field \( \vec{B} \). Find the value of work done by \( \vec{B} \).

A wire of length \( 314 \, \text{cm} \) carrying current of \( 14 \text{A} \) is bent to form a circle. The magnetic moment of the coil is __________ \( \text{A}\!\!-\!\!\text{m}^2 \). [Given \( \pi = 3.14 \) ]

A proton moving with a constant velocity passes through a region of space without any change in its velocity. If $\vec{E}$ and $\vec{B}$ represent the electric and magnetic fields respectively, then the region of space may have :
(A) $\quad E = 0, B = 0$
(B) $\quad E = 0, B \neq 0$
(C) $\quad E \neq 0, B = 0$
(D) $\quad E \neq 0, B \neq 0$

Choose the most appropriate answer from the options given below :

The magnetic moments associated with two closely wound circular coils \( A \) and \( B \) of radius \( r_{A} = 10 \, \text{cm} \) and \( r_{B} = 20 \, \text{cm} \) respectively are equal if: (Where \( N_{A}, I_{A} \) and \( N_{B}, I_{B} \) are number of turn and current of \( A \) and \( B \) respectively)

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): In an uniform magnetic field, speed and energy remains the same for a moving charged particle.
Reason (R): Moving charged particle experiences magnetic force perpendicular to its direction of motion.

A long solenoid with \( 1000 \) turns/m has a core material with relative permeability \( 500 \) and volume \( 10^{3} \, \text{cm}^{3} \). If the core material is replaced by another material having relative permeability of \( 750 \) with same volume maintaining same current of \( 0.75 \, \text{A} \) in the solenoid, the fractional change in the magnetic moment of the core would be approximately \( \left( \frac{x}{499} \right) \). Find the value of \( x \).

A long solenoid is formed by winding \( 70 \) turns \( \text{cm}^{-1} \). If \( 2.0 \, \text{A} \) current flows, then the magnetic field produced inside the solenoid is __________ \( (\mu = 4\pi \times 10^{-7} \, \text{TmA}^{-1}) \)

Two charged particles, having same kinetic energy are allowed to pass through a uniform magnetic field perpendicular to the direction of motion. If the ratio of radii of their circular paths is \( 6 : 5 \) and their respective masses ratio is \( 9 : 4 \). Then, the ratio of their charges will be :

A particle of charge \( 1.6 \, \mu\text{C} \) and mass \( 16 \, \mu\text{g} \) is present in a strong magnetic field of \( 6.28 \, \text{T} \). The particle is then fired perpendicular to magnetic field. The time required for the particle to return to original location for the first time is ________ \( \text{s} \). \( (\pi = 3.14) \)

Two long parallel wires carrying currents \( 8 \, \text{A} \) and \( 15 \, \text{A} \) in opposite directions are placed at a distance of \( 7 \, \text{cm} \) from each other. A point \( P \) is at equidistant from both the wires such that the lines joining the point \( P \) to the wires are perpendicular to each other. The magnitude of magnetic field at \( P \) is _____ \( \times 10^{-6} \, \text{T} \). \( \left( \text{Given} : \sqrt{2} = 1.4 \right) \).

The current of \( 5\,\text{A} \) flows in a square loop of side \( 1\,\text{m} \) is placed in air. The magnetic field at the centre of the loop is \( X\sqrt{2} \times 10^{-7}\ \text{T} \). The value of \( X \) is ______.

Two concentric circular loops of radii \( r_1 = 30\,\text{cm} \) and \( r_2 = 50\,\text{cm} \) are placed in X–Y plane as shown in the figure. A current \( I = 7\,\text{A} \) is flowing through them in the direction as shown in figure. The net magnetic moment of this system of two circular loops is approximately:

Two insulated circular loop A and B of radius \(' a' \) carrying a current of \( 'I' \) in the anti clockwise direction as shown in the fig. The magnitude of the magnetic induction at the centre will be:

A triangular shaped wire carrying \( 10 \, \text{A} \) current is placed in a uniform magnetic field of \( 0.5 \, \text{T} \), as shown in figure. The magnetic force on segment \( CD \) is:
(Given \( BC = CD = BD = 5 \, \text{cm} \))

A rectangular loop of sides \( 12 \, \text{cm} \) and \( 5 \, \text{cm} \), with its sides parallel to the \( x \)-axis and \( y \)-axis respectively, moves with a velocity of \( 5 \, \text{cm/s} \) in the positive \( x \) axis direction, in a space containing a variable magnetic field in the positive \( z \) direction. The field has a gradient of \( 10^{-3} \, \text{T} / \text{cm} \) along the negative \( x \) direction and it is decreasing with time at the rate of \( 10^{-3} \, \text{T} / \text{s} \). If the resistance of the loop is \( 6 \, \text{m}\Omega \), the power dissipated by the loop as heat is ______ \( \times 10^{-9} \, \text{W} \).

An infinitely long hollow conducting cylinder with radius \( R \) carries a uniform current along its surface. Choose the correct representation of magnetic field \( (B) \) as a function of radial distance \( (r) \) from the axis of cylinder.

A compass needle oscillates \( 20 \) times per minute at a place where the dip is \( 30^{\circ} \) and \( 30 \) times per minute where the dip is \( 60^{\circ} \). The ratio of total magnetic field due to the earth at two places respectively is \( \dfrac{4}{\sqrt{x}} \). The value of \( x \) is:

Two \( 10 \, \text{cm} \) long, straight wires, each carrying a current of \( 5 \, \text{A} \) are kept parallel to each other. If each wire experienced a force of \( 10^{-5} \, \text{N} \), then separation between the wires is _______ \( \text{cm} \).

A proton and an alpha particle of the same velocity enter in a uniform magnetic field which is acting perpendicular to their direction of motion. The ratio of the radii of the circular paths described by the alpha particle and proton is:

A uniform conducting wire of length \( 24a \), and resistance \( R \) is wound up as a current carrying coil in the shape of an equilateral triangle of side \( a \) and then in the form of a square of side \( a \). The coil is connected to a voltage source \( V_0 \). The ratio of magnetic moment of the coils of equilateral triangle to that for square is \( 1 : \sqrt{y} \) where \( y \) is __________.

A solenoid of \( 1000 \) turns per meter has a core with relative permeability \( 500 \). Insulated windings of the solenoid carry an electric current of \( 5 \, \text{A} \). The magnetic flux density produced by the second is: (permeability of free space \( = 4\pi \times 10^{-7} \, \text{H} / \text{m} \) )

A coil in the shape of an equilateral triangle of side \( 10 \, \text{cm} \) lies in a vertical plane between the pole pieces of permanent magnet producing a horizontal magnetic field \( 20 \, \text{mT} \). The torque acting on the coil when a current of \( 0.2 \, \text{A} \) is passed through it and its plane becomes parallel to the magnetic field will be \( \sqrt{x}\times 10^{-5} \, \text{Nm} \). The value of \( x \) is ____________.

A current carrying circular loop of radius \( 2 \ \text{cm} \) with unit normal \( \hat{n}=\dfrac{\hat{k}+\hat{i}}{\sqrt{2}} \) is placed in a magnetic field, \( \vec{B}=B_{0}(3\hat{i}+2\hat{k}) \). If \( B_{0}=4\times 10^{-3} \ \text{T} \) and current \( I=200\sqrt{2} \ \text{A} \), the torque experienced by the loop is ______ \( \text{Wb}\!\cdot\!\text{A} \).
[\( \pi = 3.14 \)]

Two identical long current carrying wires are bent into the shapes shown in the following figures. If the magnitude of magnetic fields at the centres \( P \) and \( Q \) of a semicircular arc are \( B_{1} \) and \( B_{2} \) respectively, then the ratio \( \dfrac{B_{1}}{B_{2}} \) is____.

Consider a long straight wire of a circular cross-section (radius \( a \)) carrying a steady current \( I \). The current is uniformly distributed across this cross-section. The distances from the centre of the wire's cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be:

The magnetic intensity at the centre of a long current carrying solenoid is found to be \( 1.6 \times 10^3 \, \text{A m}^{-1} \). If the number of turns is \( 8 \) per cm, then the current flowing through the solenoid is ________ \( \text{A} \).

A single ionized magnesium atom \( (A = 20) \) ion is accelerated to kinetic energy \( 5 \, \text{keV} \), and is projected perpendicularly into a magnetic field \( B \) of the magnitude \( 0.5 \, \text{T} \). The radius of path formed will be _____ \( \text{cm} \).

A velocity selector consists of electric field \( \vec{E} = E\,\hat{k} \) and magnetic field \( \vec{B} = B\,\hat{\jmath} \) with \( B = 12\,\text{mT} \). The value of \( E \) required for an electron of energy \( 728\,\text{eV} \) moving along the positive x-axis to pass undeflected is:

A charge particle is moving in a uniform magnetic field \( (2\hat{i} + 3\hat{j}) \, \text{T} \). If it has an acceleration of \( (\alpha \hat{i} - 4\hat{j}) \, \text{m}/\text{s}^2 \), then the value of \( \alpha \) will be:

The magnetic field at the centre of current carrying circular loop is \( B_1 \). The magnetic field at a distance of \( \sqrt{3} \) times radius of the given circular loop from the centre on its axis is \( B_2 \). The value of \( {B_1}/{B_2} \) will be:

Two parallel, long wires are kept \( 0.20 \, \text{m} \) apart in vacuum, each carrying current of \( x \, \text{A} \) in the same direction. If the force of attraction per meter of each wire is \( 2 \times 10^{-6} \, \text{N} \), then the value of \( x \) is approximately:

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JEE Main 2023 13th April Morning Shift

The elastic potential energy stored in a steel wire of length \( 20 \, \text{m} \) stretched through \( 2 \, \text{cm} \) is \( 80 \, \text{J} \). The cross sectional area of the wire is ________ \( \text{mm}^2 \). (given \( y = 2.0 \times 10^{11} \, \text{Nm}^{-2} \))