Biot - Savart Law

Can electric current have any effect on a Magnetic needle ? Are Electric and Magnetic Fields related ? Can Electric Current induce Magnetic Field ?

B - Straight Wire

Derivation of Magnetic Field due to a Finite, Semi-infinite and Infinite straight current carrying wire. Is there any similarity between the Electric field due to a straight line of charge and Magnetic field due to a straight current carrying wire ?

Q1

Four parallel conductors, carrying equal currents, are perpendicular to the screen and lie on the four corners of a square. In two conductors, the current is flowing into the screen, and in the other two, out of the screen. What should be the direction of current in each wire to produce a resultant magnetic field at the center of square in the direction as shown in figure ?
Calculate the magnitude of the magnetic field at the centre of the square, for the above condition.

Q2

A current i flows in a circuit shaped like an isosceles trapezium. Find the magnetic field at a point P located in the plane of the trapezium and at a distance d from the midpoint of smaller side.

Q3

Find the magnetic field at the points shown in figures below.

B - Moving Charge

Does an Electric charge create a Magnetic Field ? Why does a Magnetic Field exert force on an Electric charge ?

Q4

A point charge q is moving at a constant velocity v in the +x direction. At the instant when the point charge is at the origin, what is the magnetic field at the following points

a) (d, 0, 0)
b) (0, d, 0)
c) (0, 0, d)
d) (d, d, d)

Field - Arc

Magnetic Field due to a current carrying Arc and Circular current loop. Is there any similarity between the Electric field due to a circular loop of charge and Magnetic field due to a current carrying circular loop ?

Q5

Find the magnitude and direction of magnetic field due to each of the semi-infinite wires and the semi-circular wire at the center of semi-circle shown in the figures below.
Current in wires is i, and radius of semi-circle is R.

Q6

Two circular current loops are shown in the figure. Outer loop has radius R1 and current i1. Inner loop has radius R2 and current i2.
a) Find the expression of net magnetic field at the center of loops.
b) For what condition will the net field be zero ?
c) What will be the net field at the center in the second case.
d) For the third figure, if radius and current of loops is R and i, what is the net magnetic field at the center.

Q7

Calculate the magnetic field at point marked in each of the following cases.

Q8

Find the magnetic field at the point marked in the figures below.
Current in loops is i, and bigger radius is R1 and smaller radius is R2

Q9

Each of the figures below, show a set of current carrying straight wires and arcs. Find the magnetic field at the points marked in each case.

Q10

A steady current is flowing in a circular coil of radius R, made up of a thin conducting wire. Now, a circular loop of radius R/n is made from the same wire without changing its length, by unfolding and refolding the loop, and the same current is passed through it. Ratio of magnetic field at the center in two cases is
a. n1/2
b. n
c. n2
d. none of thesea

Q11

A current i flows through a wire shaped as regular polygon of n sides which can be inscribed in a circle of radius R.
What is the magnetic field at the centre of polygon ?
What will be magnetic field at the center of
a) an Equilateral triangular loop b) a square loop

Q12

Figures show a set of 6 infinite wires carrying the same current and placed at the same distance from the point in center. For which of the cases is it possible to have a zero magnetic field at the point in center.

B - Axis of Loop

Magnetic Field on the Axis of a Circular current loop.

Q13

Find the magnetic field at the points marked in each figure.
Current in same direction
Current in opp direction
If current in bigger loop is i1 and that in smaller loop is i2, for what ratio (and direction) of current in loops, will the magnetic field at the marked point be zero ?
e) What is the magnitude and direction of field at origin ?

B in Solenoid

Mathematical derivation of Magnetic Field due to a Solenoid

Q14

A solenoid with length L and radius R (L >> R) has 2 layers of winding of N turns each. A wire of length R/4 and mass m lies inside the solenoid near its centre normal to its axis. Both the wire and the axis of the solenoid are in the horizontal plane. The wire is connected through two leads parallel to the axis of the solenoid to an external battery which supplied a current i2 in the wire. What value of current ( with appropriate sense of direction ) in the solenoid can support the weight of the wire ?

B - Sheet

Mathematical derivation of Magnetic Field due to a Sheet of current.

Q15

Four infinite thin current carrying sheets are placed as shown in the figure. Direction of current and current density has also been marked in the fig.
What is the resultant magnetic field in all regions, inside and outside the set of sheets ?

B - Dipole

Mathematical derivation of Magnetic Field due to a Magnet or Magnetic Dipole. Is there any similarity between the Electric field of an Electric dipole and Magnetic field of a Magnetic dipole ?

Q16

A small current-carrying coil is located at the axis of a big circular loop, as shown in the figure. Find the magnitude and direction of force acting on the smaller loop. Given that a << R

Q17

A small current carrying loop with current i is placed in the plane of the screen as shown. Another semicircular loop having current io is placed concentrically in the same plane as that of the small loop, the radius of semicircular loop being R ( R >> a ).
Find the force acting on the bigger ring.

Ampere\'\'s Law

Is Ampere?s Law for electric current and Magnetic Field equivalent to Gauss?s Law for Electric charges and Electric Flux ?

Q18

A long straight wire placed along the z-axis, carries a current i. The is computed along the straight line PQ, P and Q are equidistant from the origin. What is the value of

Infinite Wire

Using Ampere?s Law to find the Magnetic field due to an infinite current carrying wire.

Q19

a) A current i flows along the length of an infinitely long, straight thin pipe. If distribution of current is uniform over the cross-section of pipe, what is the expression of magnetic field in space ? b) A current i is uniformly distributed over the cross section of an infinitely long, straight thick pipe of inner radius Ri and outer radius Ro. What is the expression of magnetic field in space ?

Solid Cylinder

Using Ampere?s Law to find the Magnetic field due to an infinite current carrying Solid cylinder.

Q20

Graph shown in figure is the plot of the magnitude of the magnetic field inside and outside four long wires a, b, c, and d, as functions of distance r from the axis. Current in the wires is uniformly distributed across the cross sections of the wires.
a) Which wire has the greatest radius ?
b) Which wire has the greatest current ?
c) Which wire has the greatest current density ?

Infinite Sheet

Using Ampere?s Law to find the Magnetic field due to an infinite current carrying sheet

Solenoid

Using Ampere?s Law to find the Magnetic field due to an Ideal Solenoid

Toroid

Using Ampere?s Law to find the Magnetic field due to a current carrying Torroid

Q21

Two long parallel wires are at a distance 2d apart. They carry equal currents flowing
a) in the same direction
b) in the opposite directions
Write the expression of variation of magnetic field in the plane of wires as a function of distance from the mid-point between the wires.
What is the variation of magnitude of magnetic field B ?

Q22

Consider two long parallel wires carrying same current. Find the expression of variation of magnetic field in the plane mid-way between the wires, if current in wires a) is in the same direction b) is in opposite directions
Also draw the net magnetic field in space around the wires for both cases.

F b/w Parallel Wires

Magnetic force between a pair of infinite Parallel current carrying wires. Is it similar to the force between a pair of electric charges ?

Q23

Two long parallel wires carry currents of equal magnitude ( i ) but in opposite directions. These wires are suspended from rod PQ by chords of same length L as shown in figure (there is 1 set of chords per unit length of wire). The mass per unit length of the wires is l. In equilibrium, the chords make an angle q with the vertical. Determine the value of q, assuming it to be small.

Q24

a) Two long, parallel current carrying wires are shown in figure. Wire 1 carries a current i1 and is held firmly in position. Wire 2 carries a current i2 and is free to slide up and down (parallel to wire 1) between a set of non-conducting and frictionless guides. If the mass per unit length of wires is l kg/m, what should be the direction and value of i2 so that the lower wire stays in equilibrium at distance d below wire 1 ?

b) Now if wire 2 is help fixed and wire 1 is free to move up and down, what should be the direction and value of i1 so that the upper wire stays in equilibrium at distance d above the lower wire ? If the upper wire is displaced by a small distance, will it execute SHM ? What will be the time period of oscillation ?

Q25

When a current is passed through a helically coiled spring, the spring has a tendency to
a) expand longitudinally
b) contract longitudinally
c) no tendency to expand or contract
d) depends on the direction of current

Magnetic Pressure

Derivation of Magnetic Pressure on a current carrying surface placed inside an external magnetic field. Is Magnetic Pressure similar to Electrostatic Pressure ?

Q26

a) What is the magnetic pressure on an infinite current carrying plane, when there is no external magnetic field in space.
b) A conducting current-carrying infinite plane is placed in a uniform external magnetic field. As a result, net magnetic field on either side of the plane is as shown in the figures. Find the direction of the current and value of current density in the plane in each case. Find the magnetic pressure acting on the plane in each case.

Q27

A system consists of two parallel planes carrying currents producing a uniform magnetic field of induction Bo between the planes. Outside the planes there is no magnetic field. Find the magnetic force acting per unit area of each plane.

ElectroMagnetism

Is there anything like a Pure Electric Field or a Pure Magnetic Field ? Are Electric and Magnetic Fields just different perceptions of a single ElectroMagnetic Field ? The answer may not be that simple !

Q28

What should be the value of the electric field strength in vacuum if the energy density due to electric field is to be the same as that due to a magnetic field of strength B in vacuum ? If the strength of Electric and Magnetic Field is equal, what is the ratio energy density due to Electric and Magnetic fields.

Newton\'\'s 3rd Law

Is Newton?s Third Law always valid ? Is Newton?s Third Law valid for Electromagnetic Forces ? Is Conservation of Momentum violated for Electromagnetic forces ?

Q29

A charged particle is moving with velocity v in x-y plane. At some moment, it produces an electric field E k, at a point at distance r from the charge. What is the magnetic field at that point at that moment ?

1

Find the magnetic field at point O.

2

Figure shows infinite number of rings each having current i in the direction shown. All the rings are concentric and lie in the same plane. The radii of rings are R, 2R, 22 R, 23 R,..... Magnetic field at the centre of the rings will be

3

An infinite wire is bent in two ways to form a circular loop. Can the magnetic field be zero at the center of loop in any of the two cases ?
A circular loop with radius R and current i1 in a clockwise direction. The center of the loop is at a distance d above a long straight wire (as shown in figure). What should be the magnitude and direction of current in the wire if the magnetic field at the center of the loop is to be zero ?

4

An infinite wire carrying current i has the configuration as shown in figure. The semi-infinite straight sections are tangent to the circle, and the circular arc subtends an angle q at the center, with all sections lying in the same plane. For what value of q will the magnetic field at the center be zero ?

5

An otherwise infinite, straight wire is bent to form loops of radii a and b as shown in figures. Find the ratio a/b for the magnetic field at the common centre to be zero

6

A battery is connected between two points on the circumference of a uniform conduction ring of radius r and resistance R. One of the arcs subtends an angle q at the center. What is the value of magnetic field at the centre of ring ?
A battery is connected between three points on the circumference of a uniform conduction ring of radius r and resistance R. Angles subtended by arcs are as shown in the figure. What is the value of magnetic field at the centre of ring ?

7

Figure shows 5 symmetric circuits with equal resistance of equal length of wires.
Find the magnetic field at the center of each circuit.

8

A point charge is moving in a circle with constant angular speed w. Magnetic field produced by the charge at a fixed point P on the axis of the circle is
a)constant in magnitude only b) constant in direction only c) constant in both direction and magnitude d) constant neither in magnitude nor in direction. Point A lies in the same plane of circle at a distance 2R from the center. If the minimum time interval between two successive times at which magnetic field at point A is zero is ?t?, then angular speed w in terms of t, is a) 2?/t b) 2?/3t c) ?/3t d) ?/t

9

There are two parallel long wires as shown in figure. Consider points X, Y and Z on the line perpendicular to both the wires and also in the plane of wires. The distances are mentioned. Find
a) Magnetic Field at points X, Y and Z
b) position of points on line XYZ where B is 0.

10

Two infinite wire carrying currents i1 and i2 are lying along x- and y- axes, as shown.
Locus of points where B is zero is ___________

11

i) Two very long straight parallel wires carry equal steady currents. Distance between the wires is d. At a certain instant of time, a point charge q is at a point equidistant from the two wires in the plane of the wires. Its instantaneous velocity v is parallel to the wires. What is the magnitude of the force due to the magnetic field acting on the charge at this instant, if a) current in wires is in the opposite direction b) current in wires is in the same direction
ii) Two infinite long wires, each carrying equal current i are lying along x-and y-axis, respectively. A charged particle, having a charge q and mass m, is projected with a velocity u along the straight line y = x.
What is the path traced by the particle ? (neglect gravity)

12

A long, thin circular pipe, with radius R, carries a current i ( out of the screen as shown in figure ). A wire runs parallel to the pipe at a distance 3R from centre to centre and has the same current i. Draw the net field inside and outside the pipe, if current in wire
a) is coming out of the screen b) is going into the screen Calculate the magnitude and direction of the current in the wire that would cause the resultant magnetic field at the point P to have the same magnitude, but the opposite direction, as that of the resultant field at the centre of the pipe.

13

A thick cylindrical wire of radius R has a uniform distributed current flowing through it. ( current density = J ). A cylindrical cavity of radius r whose centre lies at distance a from the center of wire, is removed from the wire. a) Find the expression of magnetic field inside the spherical cavity. b) Find the magnetic field at the axis of wire.

14

A square loop of wire of side a, carries a current i.
a) Find the magnitude of magnetic field at a point on the axis of the loop at a distance z from the centre.
b) What is the magnetic field at the center of square loop ?
c) What is the magnetic field at a far-off point on the axis of loop ?

15

Find resultant magnetic field at point O in the figure.
The loop is now folded along the line marked in figure, such that the left half of loop is perpendicular to the right half. What is the resultant magnetic field at point O ?

16

A current carrying wire loop is placed in the x-y plane as shown in figure
a) If a particle with charge +Q and mass m is placed at the centre and given a velocity v along PO, what is the force acting on it at this moment ?
b) If a uniform external magnetic field B is applied along the horizontal direction, find the force and the torque acting on the loop due to this field.

17

Two long parallel wires carrying current 5 A and i A in the same direction (directed into the plane of the paper) are held at P and Q respectively such that they are perpendicular to the plane of paper. The point P and Q are located at a distance of 5m and 2m respectively from a collinear point R
a) An electron moving with a velocity of 4 ? 105 m/s along the positive x direction experiences a force of magnitude 4 ? 10-20 N at the point R. Find the value of i.
b) Find all the position at which a third long parallel wire carrying a current of magnitude 2.5 A may be placed, so that the magnetic field at R zero.

18

There are three long, straight, parallel wires spaced at equal distance d apart. The outer wires, fixed in their place, carry current i out of the screen, as shown in figure. Middle wires has the same magnitude of current i. It is free to slide and can be displaced slightly either along the horizontal direction or vertical direction.
In what case does the wire execute SHM , if current in middle wire,
a) is out of screen b) into the screen

19

There is a long solenoid with length = L, radius = R, number of turns per unit = n and carrying current i (assume that the field inside the solenoid is uniform).
A particle with charge q and mass m, enters the solenoid from one of the sides with velocity v and making some angle with the magnetic field.
a) Where should the particle enter the solenoid and in which direction should the particle be moving so that the number of turns taken by the particle inside the solenoid is maximum ? (ignore the case of circular path which will mean infinite number of turns)
b) What is the value of the maximum number of turns taken by the particle ?

20

A square loop carrying a current is located in the same plane as along straight wire carrying a current il . The loop side is a. Axis of loop passing through its center and parallel to the wire is at a distance iw from the wire. Find the force and torque acting on the loop.
How much work is required to be done to rotate the loop by 180o wrt the marked axis ?

21

A long straight wire is coplanar with a current carrying circular loop of radius R as shown in figure.
a) Is force between wire and loop attractive or repulsive ?
b) Calculate force of attraction between the wire and loop.

22

A straight segment of current of length L is placed along the x - axis. Two infinitely long straight wires, each extending from z = - ? to + ?, are fixed at y = -d and y = +d respectively, as shown in the figure. Find the expression for the force acting on the segment of length L.

23

Figure shows an infinite current wire (current going into the screen), and a current carrying loop.
a) What is the net force on the loop ?
b) What is the net torque (magnitude and direction) on the loop ?

24

Figure shows a circuit which is in the horizontal plane. The circuit consists of eight alternating arcs and each subtends the same angle at the centre. There is an infinite wire lying on the axis of circuit and perpendicular to it.
a) Find the magnetic field produced by this circuit at the centre.
b) What is the magnetic force on the wire ? on the circuit ?
c) What is the torque acting on the circuit ?

25

A long straight current carrying wire is placed in the plane of a current carrying ribbon as shown in the figure. Find the force per unit length between them

26

A current i flows in a long straight conductor whose cross section has the form of a thin half ring of radius R. The same current (and in same direction) flows in a thin conductor located on the axis of the first conductor. Find the magnetic force per unit length between the conductors.

27

A current i flows in a long thin walled cylinder of radius R. What pressure do the walls of the cylinder experience? Does the cylinder tend to expand or contract radially? What pressure does the lateral surface of a long straight solenoid with n turns per unit length experience when a current i flows through it? Does the solenoid tend to expand or contract radially?

28

A coaxial cable is made of two conductors. The inner conductor is solid and is of radius R1 and the outer conductor is hollow with inner radius R2 and outer radius R3. Plot the variation of magnetic field with distance from the axis,
a) if current in both pipes is of equal magnitude and in same direction
b) if current in both pipes is of equal magnitude and in opposite
direction.

29

A charged ring of radius R having uniformly distributed charge q is rotating with angular velocity w about its axis. Find the magnetic field
a) at the center of ring
b) at a point on its axis, lying at a distance z

A positively charged disk with radius R and uniform charge distribution s is rotating as shown in Figure. Find
a) the magnetic field at the center of disc
b) the magnetic field at a point on its axis, lying at a distance z
c) the direction of magnetic field at point A in the plane of the disk

30

a) A hollow spherical shell of radius R, and uniform surface charge density s, rotates about the axis passing through its centre at an angular velocity w. Find the magnetic induction at the centre of the rotating spherical shell.

b) A solid sphere of radius R, and uniform volume charge density r, rotates about the axis passing through its centre at an angular velocity w. Find the magnetic induction at the centre of the rotating sphere.

31

A coil having N turns is wound tightly in the form of spiral with inner and outer radii a and b respectively. When a current i passes through the coil, what is the magnetic field at the center ?

32

A current i flows in a circular loop. Find the integral along the axis of the loop. Explain the result obtained.

33

Current density in a wire of radius R varies with r according to Kr2, where K is a constant and r is the distance from the axis of the wire. Find the magnetic field as a function of r, both inside and outside the wire.

34

A long straight wire carries a current i. A particle having a positive charge q and mass m, kept at distance xo from the wire is projected towards it with speed v. Find the closest distance of approach of charged particle to the wire.
If the particle was projected in the opposite direction, what will be its maximum distance from the wire before it turns back ?

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