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QUESTION: 1

Which one of the following can represent the magnetic field outside the wire of radius a earning an arbitrary current distribution in its volume:

Solution:

Any vector will represent the magnetic field if the divergence of the vector field is zero. The curl of magnetic field outside the wire must be zero, i.e.,

Only option (d) satisfies tins conditions.

QUESTION: 2

Suppose four infinitely long wires passing through the corners of a square of side 2a centre at origin in x-y plane as shown in figure. The magnetic tield at origin is

Solution:

The magnetic field at the origin due to wire (I) is

Therefore, the resultant magnetic field at origin

QUESTION: 3

A long wire of radius R carries a current I with a volume current density (where α is a constant and s is the perpendicular distance from the axis). The value of α is

Solution:

Given: Current density

Tlierefore, the total current passing through the wire is

QUESTION: 4

A long hollow coaxial wire lias inner radius a and outer radius b. Uniform current I flows along its inner surface and return through the outer surface as shown in figure. The magnitude vector potential in different regions are

Solution:

According to Ampere's law.

QUESTION: 5

A conducting wire carrying current I_{0} is bent in the shape of a loop, as shown in figure. The magnetic field at Unseen point O is given by

Solution:

The magnetic field at O is due to curve portion + straight portion.

The magnetic field due to curve portion is

The magnetic field due to straight portion is

Therefore, the resultant magnetic field at O is

QUESTION: 6

A vector potntial at a point is given by The current density corresponding to the vector potential is

Solution:

Given magnetic vector potential

Therefore, the magnetic field.

Therefore, the current density

QUESTION: 7

Three infinitely-long conductors carrying currents I_{1}, I_{2}, and I_{3} lie perpendicular to the plane of the paper as shown below. If the value of integral for the loops C_{1}, C_{2}, and C_{3} are 2μ_{0}, 4_{μ}0 and μ_{0} in the units of N/A, respectively, then

Solution:

According to Ampere’s law,

Solving the above three equation, we can write

I_{2 }= 5, I_{1 }= -3, I_{3 }= -1

QUESTION: 8

If a point dipole of the dipole moment is kept in external magnetic field The force acting on the dipole is given by

Solution:

We have magnetic dipole,

And magnetic field,

Therefore, the magnetostatics energy,

Therefore, the force acting on the dipole.

*Answer can only contain numeric values

QUESTION: 9

Consider two infinitely Ions wires parallel to the z-axis passing through the point and One is carrying current I_{0} along positive z-direction of other one is carrying current 3I_{0}, along negative z-direetion. The value of on closed curve (x - 1)^{2} + y^{2} = 16 along clockwise direction is _____ μ_{0}I_{0}.

(Answer should be integer).

Solution:

According to Ampere’s law,

*Answer can only contain numeric values

QUESTION: 10

An election is revolving around the nucleus in circular orbit of radius a_{0} in a hydrogen atom. If electron is revolving with speed v_{0} , the magnetic field at nucleus due to orbital motion of electron is . (Round off to two decimal places)

Solution:

Current flow due motion of electron in circular orbit is

Therefore, the magnetic field at the centre of circular loop is

QUESTION: 11

Two infintely long current carrying wire are placed along x and y axis and crossing through the origin as shown in figure. The magnetic field at (2, 3,4) will be

Solution:

The magnetic field at (2,3,4) due to wire (I) is

The magnetic field at (2,3,4) due to wire (II) is

Therefore, the resultant magnetic field at (2,3,4) is

QUESTION: 12

Two large parallel plates move with a constant speed v in the positive y direction as shown in figure. If both the plates have surface charge density σ, the magnetic force per unit area on the lower plate is

Solution:

The current density on the moving plate is given by

Therefore, the magnetic field on the lower plate is

Therefore, the magnetic force on the lower plate per unit area is

QUESTION: 13

All infinitely long wire carrying current I_{0} is placed (crossing through the origin) along x-axis and another wire of length L carrying current I_{0} is placed mx-y plane, as shown in figure below. Thex-component of magnetic force acting on the wire is given by

Solution:

Consider an arbitrary points on the finite wire (x, y).

The magnetic field at (x, y) is

Therefore, the magnetic force on the elementary length,

Therefore, the x component of magnetic force is

QUESTION: 14

The regions of space z < 0 and z > 0 are filled with free space and material having permeability 2_{μ}0 respectively. The magnetic field m the region and the magnitude of magnetic field in region z > 0 is 6T. The angle between magnetic field and normal to the mterface in region z < 0 is given by

Solution:

According to boundary condition, B_{ln} = B_{2n}

⇒ B_{2n} = 2

∴ The angle between magnetic field and normal to the interface in region z < 0 is

QUESTION: 15

Two magnetic dipoles of magnitude m, one is placed at origin and the otherone is displaced from P to Q keeping the direction fixed (Assume that P and Q both are in x-y plane). The required work done to displaced the magnetic dipole from P to Q is

Solution:

The mutual magnetostatics energy when magnetic dipole at P is

Tlie mutual magnetostatics energy when magnetic dipole at Q is

Therefore. the required work done is

W = change of magnetostatic energy

*Answer can only contain numeric values

QUESTION: 16

A ring of radius R carries a linear charge density λ. It is rotating about an axis crossing through its centre and perpendicular to its plane with an angular speed ω. The magnetic field on its axis at a distance from the centre is ________ μ_{0}λω. (Round off to two decimal places)

Solution:

Current due to rotation of charge carrying ring is given by I = ωλR.

The magnetic field on the axis of a loop is

*Answer can only contain numeric values

QUESTION: 17

A rigid square loop ABCD of side L is situated at a distance L from an infinite straight wire, with the side AB parallel to the infinite wire, as shown in the figure below. The infinite wire, which is in the plane of the loop, carries a current I from right to left. The value of through the square loop is _____________ (Round off to two decimal places)

Solution:

*Answer can only contain numeric values

QUESTION: 18

The volume current density through a long cylindrical conductor is given to be where, R is radius of cylinder and r is tlie distance of some point from tlie axis of cylinder and J_{0} is a constant. The value of r at which magnetic field maximum is ___________________ R. (Round off to two decimal places)

Solution:

According to Ampere's law,

For extrema value of B

Now, B will be maximum for

*Answer can only contain numeric values

QUESTION: 19

A long hollow cylinder of radius R carrying uniform surface charge density σ. It is rotating about its axis with angular speed ω. A ring of radius (r = R/2) is placed eo-axially with cylinder. The value of ton the ring is ______ πμ_{0}ωσR^{3} (Round off to two decimal places)

Solution:

Given: The surface current density σ. Now, when it will rotate about its axis there will be current flow along and we can treat it as solenoid.

Therefore, the magnetic field inside the cylinder

*Answer can only contain numeric values

QUESTION: 20

A long solid conducting cylinder of radius R carrying volume current along its axis. The magnetic field inside the cylinder is given by (where J_{0} is a constant and r is the distance from the it axis). The magnetic field at a distance r = 2R from it axis is______

(Upto two decimal places)

Solution:

We have magnetic field,

Therefore, the current density corresponding to the magnetic field,

Therefore, the magnetic field at outside the cylinder

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