Moving charges and magnetism

 1. Magnetic Effect of Current

A magnetic field is associated with an electric current flowing through a metallic wire. This is called magnetic effect of current. On the other hand, a stationary electron produces electric field only.

2. Source and Units of Magnetic Field

Oersted’s Experiment: A Danish physicist, Hans Christian Oersted, in 1820, demonstrated that a magnetic needle is deflected by a current carrying wire. He concluded that the magnetic field is caused by current elements (or moving charges). The unit of magnetic field strength in SI system is tesla (T) or weber/metre2 (Wb m–2) or newton/ampere-metre (N A–1 m–1).

In CGS system, the unit of magnetic field is gauss (G).

1T=104 G

3. Biot-Savart Law

It states that the magnetic field strength dB−→produced due to a current element (of current I and length dl) at a point having position vector r⃗ relative to current element is

dB−→−=μ04πIdl→×r⃗ r3

where µ0 is permeability of free space. Its value is

µ0 = 4π ×10–7 Wb/A-m.

The magnitude of magnetic field is

dB=μ04πIdlsinθr2

where θ is the angle between current element Idland position vector r⃗ as shown in the figure.

The direction of magnetic field dB−→is perpendicular to the plane containing Idl−→ and r→.

4. Magnetic Field due to a Circular Coil

The magnetic field due to current carrying circular coil of N-turns, radius a, carrying current I at a distance x from the centre of coil is

B=μ0NIa22(a2+x2)3/2 along the axis.

At centre, x = 0

∴ Bc=μ0NI2a

The direction of magnetic field at the centre is perpendicular to the plane of the coil.

In general the field produced by a circular arc subtending an angle θ at centre is

BC=μ0I2a.θ2π (θ in radian)

5. Ampere’s Circuital Law

It states that the line integral of magnetic field B⃗  along a closed path is equal to µ0-times the current (I) passing through the closed path.

∮B⃗ .dl→=μ0I

6. Magnetic Field due to a Straight Conductor Carrying a Current using

Biot-Savart Law

The magnetic field due to a straight current carrying wire of finite length at a point is

B=μ0I4πR(sinφ1+sinφ2)

where R is the perpendicular distance of the point from the conductor.

The direction of magnetic field is given by right hand grip rule.

Special cases: (i) If the wire is infinitely long, then φ1 = π/2, φ2 = π/2

B=μ0I2πR

(ii) If point is near one end of a long wire, (φ1=π2,φ2=0), then

B=μ0I4πR

7. Magnetic Field due to a Current Carrying Solenoid

At the axis of a long solenoid, carrying a current I

B=µ0nI

where n = number of turns per unit length.

Magnetic field at one end of solenoid Bend=μ0nI2

The polarity of any end is determined by using Ampere’s right hand rule.

8. Force on a Moving Charged Particle in Magnetic Field

The force on a charged particle moving with velocity v⃗ in a uniform magnetic field B⃗ is given by

F⃗ m=q(v⃗ ×B⃗ )=qvBsinθ

This is known as Lorentz force.

The direction of this force is determined by using Fleming’s left hand rule.

The direction of this force is perpendicular to both v⃗  and B⃗ ,

When v⃗  is parallel to B⃗ , then F⃗ m=0

When v⃗  is perpendicular to B⃗ , then F⃗ m is maximum, i.e., Fm=qvB.

9. Force on a Charged Particle in Simultaneous Electric and Magnetic Fields

The total force on a charged particle moving in simultaneous electric field E⃗  and magnetic field B−→is given by

F⃗ =q(E⃗ +v⃗ ×B⃗ )

This is called Lorentz force equation.

10. Path of Charged Particle in a Uniform Magnetic Field

(i) If v⃗  is parallel to the direction of B,−→− then magnetic force = zero. So the path of particle is an undeflected straight line.

(ii) If v⃗  is perpendicular to B⃗ , then magnetic field provides a force whose direction is perpendicular to both v⃗ andB⃗  and the particle follows a circular path. The radius r of path is given by

mv2r=qvB⇒r=mvqB

If K is kinetic energy of a particle, then P=mv=2mK−−−−−√

r=2mK√qB

If V is accelerating potential in volt, K = qV

∴r=2mqV√qB=1B2mVq−−−−√

Time period of revolution is T=2πmqB

(iii) If a particle’s velocity v⃗  is oblique to magnetic field B,−→− then the particle follows a helical path of radius, r=mvsinθqB=mv⊥qB

Time period, T=2πmqB

and pitch, P=v∣∣T=vcosθ2πmqB

where v∣∣ is a component of velocity parallel to the direction of magnetic field.

11. Velocity Filter

If electric and magnetic fields are mutually perpendicular and a charged particle enters this region with velocity v⃗  which is perpendicular to both electric and magnetic fields, then it may happen that the electric and magnetic forces are equal and opposite and charged particle with given velocity v remain undeflected in both fields. In such a condition

qE=qvB⇒v=EB

This arrangement is called velocity filter or velocity selector.

12. Magnetic Force on a Current Carrying Conductor of Length l⃗  is given by

F⃗ m=I(l⃗ ×B⃗ )

Magnitude of force is

Fm = IlB sin θ

Direction of force F⃗  is normal to l⃗  and B⃗  given by Fleming’s Left Hand Rule. If θ=0(.i.e.,l⃗ isparalleltoB⃗ ), then magnetic force is zero.

13. Force between Parallel Current Carrying Conductors

Two parallel current carrying conductors attract while antiparallel current carrying conductors repel. The magnetic force per unit length on either current carrying conductor at separation ‘r’ is given by

Fl=μ0I1I22πrnewton/metre

=2×10–7I1I2r

Its unit is newton/metre abbreviated as N/m.

n Definition of ampere in SI System

1 ampere is the current which when flowing in each of the two parallel wires in vacuum at separation of 1 m from each other exert a force of

μ02π=2×10–7N/m on each other.

14. Torque Experienced by a Current Loop (of Area A⃗ ) Carrying Current I in a Uniform Magnetic Field B⃗  is given by

τ⃗ =NI(A⃗ ×B)−→−=M⃗ ×B⃗ 

where M⃗ =NIA⃗  is magnetic moment of loop. The unit of magnetic moment in SI system is ampere × metre2 (Am2).

15. Potential energy of a current loop in a magnetic field

When a current loop of magnetic moment M is placed in a magnetic field, then potential energy of magnetic dipole is

U=–M⃗ .B⃗ =–MBcosθ

(i) When θ = 0, U = – MB (minimum or stable equilibrium position)

(ii) When θ = π, U = + MB (maximum or unstable equilibrium position)

(iii) When θ=π2,potential energy is zero.

16. Moving Coil Galvanometer

A moving coil galvanometer is a device used to detect flow of current in a circuit.

A moving coil galvanometer consists of a rectangular coil placed in a uniform radial magnetic field produced by cylindrical pole pieces. Torque on coil τ = NIAB where N is number of turns, A is area of coil. If C is torsional rigidity of material of suspension wire, then for deflection θ, torque τ = Cθ

∴ For equilibrium, NIAB = Cθ

⇒θ=NABCI⇒θ∝I

Clearly, deflection in galvanometer is directly proportional to current, so the scale of galvanometer is linear.

Figure of Merit of a galvanometer: The current which produces a deflection of one scale division in the galvanometer is called its figure of Merit. It is equal to Iθ=CNAB

Sensitivity of a galvanometer: Current sensitivity: It is defined as the deflection of coil per unit current flowing in it.

Sensitivity SI=(θI)=NABC

Voltage sensitivity: It is defined as the deflection of coil per unit potential difference across its ends

i.e., SV=θV=NABRg.C,

where Rg is resistance of galvanometer.

Clearly for greater sensitivity, number of turns N, area A and magnetic field strength B should be large and torsional rigidity C of suspension should be small.

17. Conversion of Galvanometer into Ammeter

A galvanometer may be converted into ammeter by using very small resistance in parallel with the galvanometer coil. The small resistance connected in parallel is called a shunt. If G is resistance of galvanometer, Ig is current in galvanometer for full scale deflection, then for conversion of galvanometer into ammeter of range I ampere, the shunt is given by

S=IgI–IgG

18. Conversion of Galvanometer into Voltmeter

A galvanometer may be converted into voltmeter by connecting high resistance (R) in series with the coil of galvanometer. If V volt is the range of voltmeter formed, then series resistance is given by

R=VIg–G