CHAPTER -4 MOVING CHARGES AND MAGNETISM

An electron and a positron are released from (0, 0, 0) and
(0, 0, 1.5R ) respectively, in a uniform magnetic field B = 0
B ˆi , each
with an equal momentum of magnitude p = e BR. Under what
conditions on the direction of momentum will the orbits be nonintersecting
circles?

A rectangular conducting loop consists of two wires on two opposite
sides of length l joined together by rods of length d. The wires are
each of the same material but with cross-sections differing by a
factor of 2. The thicker wire has a resistance R and the rods are of
low resistance, which in turn are connected to a constant voltage
source V0. The loop is placed in uniform a magnetic field B at 45°
to its plane. Find τ, the torque exerted by the magnetic field on the
loop about an axis through the centres of rods.

Do magnetic forces obey Newton’s third law. Verify for two current
elements dl1 = dl ˆi located at the origin and dl2 = dl ˆj located at
(0, R, 0). Both carry current I.

An electron enters with a velocity v = v0i into a cubical region (faces
parallel to coordinate planes) in which there are uniform electric
and magnetic fields. The orbit of the electron is found to spiral
down inside the cube in plane parallel to the x-y plane. Suggest a
configuration of fields E and B that can lead to it.

A charged particle of charge e and mass m is moving in an electric
field E and magnetic field B. Construct dimensionless quantities
and quantities of dimension [T ]–1.

A current carrying loop consists of 3 identical quarter circles of
radius R, lying in the positive quadrants of the x-y, y-z and z-x
planes with their centres at the origin, joined together. Find the
direction and magnitude of B at the origin.

Describe the motion of a charged particle in a cyclotron if the
frequency of the radio frequency (rf) field were doubled.

The magnetic force depends on v which depends on the inertial
frame of reference. Does then the magnetic force differ from inertial
frame to frame? Is it reasonable that the net acceleration has a
different value in different frames of reference?

Show that a force that does no work must be a velocity dependent
force.

Verify that the cyclotron frequency ω = eB/m has the correct
dimensions of [T]–1.

What Does a toroid consist of? Show that for and ideal toroid of closely wound turns the magnetic field.

1. Inside the toroid is constant.
2. In the open space inside an exterior to toroid is zero.

Explain how does an atom behave as a magnetic dipole. Derive an expression for the magnetic dipole moment of the atom. Also define Bohr magneton. 

Define the terms (i) magnetizing field, (ii) magnetic intensity, (iii) magnetisation, (iv) relative permeability and (v) magnetic susceptibility. Give their S I unit, if any

Derive a mathematical expression for the force per unit length acting on each of the two straight parallel metallic conductors carrying current in the same direction and kept near each other. Hence define an ampere. Why do such current carrying conductors attract each other? 

With the help of a neat and labeled diagram, explain the underlying principle, construction and working of a moving coil galvanometer. What is the function of (i) uniform radial field (ii) soft iron core in such a device? 

Derive an expression for the torque on a rectangular coil of area A, carrying a current I and placed in a magnetic field B. The angle between the direction of B and vector perpendicular to the plane of the coil is 

1. With the help of a labeled diagram, explain the principle, construction, theory and working of a cyclotron. 

Discuss the motion of a charged particle in a uniform magnetic field with initial velocity

(i)    parallel to the field, (ii) perpendicular to the magnetic field and (iii) at an arbitrary angle with the field direction. 

. A long solenoid with closely wound turns has n turns per unit of its length. A steady current I flow through this solenoid. Use Ampere’s circuital law to obtain an expression for the magnetic field at a point on its axis and close to its mid point. 

Using Biot-Savart law, deduce an expression for the magnetic field on the axis of a circular current loop. Hence obtain the expression for the magnetic field at the centre of the loop. 

What are magnetic lines of force? Give their important properties.

How will you select materials for making permanent magnets, electromagnets and cores of transformers? 

How can a galvanometer be converted into a voltmeter to read a maximum potential difference V? Discuss with related mathematical expression

How will you convert a galvanometer into an ammeter of range 0 - I amperes? What is the effective resistance of an ammeter? 

Define current sensitivity and voltage sensitivity of a galvanometer. State the factors on which the sensitivity of a moving coil galvanometer depends. 

Derive a mathematical expression for the force acting on a current carrying straight conductor kept in a magnetic field. Under what conditions is this force (i) zero and (ii) maximum?

State the factors on which the force acting on a charge moving in a magnetic field depends. Write the expression for this force. When is this force minimum and maximum? 

State Biot-Savart law for magnetic field produced at a point due to a small current element. How will you find the direction of magnetic field? 

1. 11.              A circular coil of N turns and radius R carries a current I. It is unwound and rewound to make another coil of radius R/2, current I remaining same. Calculate the ratio of the

magnetic moments of the now coil and the original coil. 

Which one of the two, an ammeter or a milliammeter, has a higher resistance and why

A circular coil of ‘N’ turns and diameter ‘d’ carries a current ‘I’. It is unwound and rewound to make another coil of diameter ‘2d’, current ‘I’ remaining the same. Calculate the ratio of the magnetic moments of the new coil and the original coil.

1. 9.                  A steady current (I1) flows through a long straight wire. Another wire carrying steady current (I2) in the same direction is kept close and parallel to the first wire. Show with the help of a diagram how the magnetic field due to the current I1 exerts a magnetic force on the second wire. Write the expression for this force. 

1. 8.                  Deduce the expression for the magnetic dipole moment of an electron orbiting around the central nucleus. 

1. 7.                  Draw magnetic field lines when a (i) diamagnetic, (ii) paramagnetic substance is placed in an external magnetic field. Which magnetic property distinguishes this behaviour of the field lines due to the two substances? 

1. 6.                  (i) Write two characteristics of a material used for making permanent magnets. (ii) Why is core of an electromagnet made of ferromagnetic materials? 

1. 4.                  A wire of length L is bent round in the form of a coil having N turns of same radius. If a steady current I flows through it in a clockwise direction, find the magnitude and direction of the magnetic field produced at its centre. 

1. 3.                  Define current sensitivity and voltage sensitivity of a galvanometer. Increasing the

current sensitivity may not necessarily increase the voltage sensitivity of a galvanometer. Justify.

1. 1.                  Define the term magnetic dipole moment of a current loop. Write the expression for the magnetic moment when an electron revolves at a speed ‘v’, around an orbit of radius ‘ r’ in hydrogen atom. 

Where on the surface of the earth is the vertical component of earth’s magnetic fieldzero?

1. 12.              Steel is preferred for making permanent magnets whereas soft iron is preferred for making electromagnets .Give one reason. 

1. 12.              Why should the spring or suspension wire in a moving coil galvanometer have low torsional constant? 

1. 10.              An electron is moving along positive x axis in the presence of uniform magnetic field along positive y axis. What is the direction of the force acting on it? 

1. 10.              Mention the two characteristic properties of the material suitable for making core of a transformer. 

Where on the surface of Earth is the angle of dip zero

1. 7.                  The permeability of a magnetic material is 0.9983. Name the type of magnetic materials it represents. 

1.                   What is the characteristic property of a diamagnetic material? 

1.                   What is the characteristic property of a diamagnetic material? 

1.                 An electron does not suffer any deflection while passing through a region of uniform magnetic field. What is the direction of the magnetic field? 

1.                 Magnetic field lines can be entirely confined within the core of a toroid, but not within a straight solenoid. Why? 

1.                   What is the direction of the force acting on a charge particle q, moving with a velocity v in a uniform magnetic field B? 

A galvanometer coil has a resistance of 15 Ω and the metre shows full scale deflection for a current of 4 mA. How will you convert the metre into an ammeter of range 0 to 6 A?

A galvanometer coil has a resistance of 12 Ω and the metre shows full scale deflection for a current of 3 mA. How will you convert the metre into a voltmeter of range 0 to 18 V?

A solenoid 60 cm long and of radius 4.0 cm has 3 layers of windings of 300 turns each. A 2.0 cm long wire of mass 2.5 g 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 supplies a current of 6.0 A in the wire. What value of current (with appropriate sense of circulation) in the windings of the solenoid can support the weight of the wire? g = 9.8 m s^-2

A circular coil of 20 turns and radius 10 cm is placed in a uniform magnetic field of 0.10 T normal to the plane of the coil. If the current in the coil is 5.0 A, what is the

(a) total torque on the coil,

(b) total force on the coil,

(c) average force on each electron in the coil due to the magnetic field?

(The coil is made of copper wire of cross-sectional area 10^-5 m², and the free electron density in copper is given to be about 10²⁹ m^-3.)

A uniform magnetic field of 3000 G is established along the positive z-direction. A rectangular loop of sides 10 cm and 5 cm carries a current of 12 A. What is the torque on the loop in the different cases shown in Fig. 4.28? What is the force on each case? Which case corresponds to stable equilibrium?

A uniform magnetic field of 1.5 T exists in a cylindrical region of radius10.0 cm, its direction parallel to the axis along east to west. A wire carrying current of 7.0 A in the north to south direction passes through this region. What is the magnitude and direction of the force on the wire if,

(a) the wire intersects the axis,

(b) the wire is turned from N-S to northeast-northwest direction,

(c) the wire in the N-S direction is lowered from the axis by a distance of 6.0 cm?

The wires which connect the battery of an automobile to its starting motor carry a current of 300 A (for a short time). What is the force per unit length between the wires if they are 70 cm long and 1.5 cm apart? Is the force attractive or repulsive?

A straight horizontal conducting rod of length 0.45 m and mass 60 g is suspended by two vertical wires at its ends. A current of 5.0 A is set up in the rod through the wires.

(a) What magnetic field should be set up normal to the conductor in order that the tension in the wires is zero?

(b) What will be the total tension in the wires if the direction of current is reversed keeping the magnetic field same as before? (Ignore the mass of the wires.) g = 9.8 m s^-2.

A magnetic field set up using Helmholtz coils (described in Exercise 4.16) is uniform in a small region and has a magnitude of 0.75 T. In the same region, a uniform electrostatic field is maintained in a direction normal to the common axis of the coils. A narrow beam of (single species) charged particles all accelerated through 15 kV enters this region in a direction perpendicular to both the axis of the coils and the electrostatic field. If the beam remains undeflected when the electrostatic field is 9.0 x 10-5 V m-1, make a simple guess as to what the beam contains. Why is the answer not unique?

An electron emitted by a heated cathode and accelerated through a potential difference of 2.0 kV, enters a region with uniform magnetic field of 0.15 T. Determine the trajectory of the electron if the field (a) is transverse to its initial velocity, (b) makes an angle of 30º with the initial velocity.

Answer the following questions:

(a) A magnetic field that varies in magnitude from point to point but has a constant direction (east to west) is set up in a chamber. A charged particle enters the chamber and travels undeflected along a straight path with constant speed. What can you say about the initial velocity of the particle?

(b) A charged particle enters an environment of a strong and non-uniform magnetic field varying from point to point both in magnitude and direction, and comes out of it following a complicated trajectory. Would its final speed equal the initial speed if it suffered no collisions with the environment?

(c) An electron travelling west to east enters a chamber having a uniform electrostatic field in north to south direction. Specify the direction in which a uniform magnetic field should be set up to prevent the electron from deflecting from its straight line path.

A toroid has a core (non-ferromagnetic) of inner radius 25 cm and outer radius 26 cm, around which 3500 turns of a wire are wound. If the current in the wire is 11 A, what is the magnetic field (a) outside the toroid, (b) inside the core of the toroid, and (c) in the empty space surrounded by the toroid.

For a circular coil of radius R and N turns carrying current I, the magnitude of the magnetic field at a point on its axis at a distance x from its centre is given by,

(a) Show that this reduces to the familiar result for field at the centre of the coil.

(b) Consider two parallel co-axial circular coils of equal radius R, and number of turns N, carrying equal currents in the same direction, and separated by a distance R. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to R, and is given by,

, approximately.

[Such an arrangement to produce a nearly uniform magnetic field over a small region is known as Helmholtz coils.]

A magnetic field of 100 G (1 G = 10-4 T) is required which is uniform in a region of linear dimension about 10 cm and area of cross-section about 10-3 m2. The maximum current-carrying capacity of a given coil of wire is 15 A and the number of turns per unit length that can be wound round a core is at most 1000 turns m-1. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic

Two concentric circular coils X and Y of radii 16 cm and 10 cm, respectively, lie in the same vertical plane containing the north to south direction. Coil X has 20 turns and carries a current of 16 A; coil Y has 25 turns and carries a current of 18 A. The sense of the current in X is anticlockwise, and clockwise in Y, for an observer looking at the coils facing west. Give the magnitude and direction of the net magnetic field due to the coils at their centre.

(a) A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of 60º with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning.

(b) Would your answer change, if the circular coil in (a) were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars are also unaltered.)

Two moving coil meters, M1 and M2 have the following particulars:

R1 = 10 Ω, N1 = 30,

A1 = 3.6 x 10-3 m2, B1 = 0.25 T

R2 = 14 Ω, N2 = 42,

A2 = 1.8 x 10-3 m2, B2 = 0.50 T

(The spring constants are identical for the two meters).

Determine the ratio of (a) current sensitivity and (b) voltage sensitivity of M2 and M1.

A square coil of side 10 cm consists of 20 turns and carries a current of 12 A. The coil is suspended vertically and the normal to the plane of the coil makes an angle of 30º with the direction of a uniform horizontal magnetic field of magnitude 0.80 T. What is the magnitude of torque experienced by the coil?

A closely wound solenoid 80 cm long has 5 layers of windings of 400 turns each. The diameter of the solenoid is 1.8 cm. If the current carried is 8.0 A, estimate the magnitude of B inside the solenoid near its centre.

Two long and parallel straight wires A and B carrying currents of 8.0 A and 5.0 A in the same direction are separated by a distance of 4.0 cm. Estimate the force on a 10 cm section of wire A.

A 3.0 cm wire carrying a current of 10 A is placed inside a solenoid perpendicular to its axis. The magnetic field inside the solenoid is given to be 0.27 T. What is the magnetic force on the wire?

What is the magnitude of magnetic force per unit length on a wire carrying a current of 8 A and making an angle of 30º with the direction of a uniform magnetic field of 0.15 T?

A horizontal overhead power line carries a current of 90 A in east to west direction. What is the magnitude and direction of the magnetic field due to the current 1.5 m below the line?

A long straight wire in the horizontal plane carries a current of 50 A in north to south direction. Give the magnitude and direction of B at a point 2.5 m east of the wire.

 
A long straight wire carries a current of 35 A. What is the magnitude of the field B at a point 20 cm from the wire?

A circular coil of wire consisting of 100 turns, each of radius 8.0 cm carries a current of 0.40 A. What is the magnitude of the magnetic field B at the centre of the coil?