Moving Charges and Magnetism

For more than a century, moving charges and magnetism, i.e., electricity, have been explored. The phenomenon observed with the alignment of a needle was the origin of the relationship between the two. It was found that the alignment of the needle is tangent to an imaginary circle with a straight wire in the centre and a plane of the circle perpendicular to the wire. However, when the current is passed, the needle’s orientation shifts. The production of a magnetic field is thought to be caused by the passage of charges. The direction of deflection of the needle is given by Ampere’s swimming rule.

Table of Contents

• Ampere’s Swimming Rule
• Magnetic Field
• Motion of a Charged Particle in a Magnetic Field
• Biot-Savart Law
• Ampere’s Circuital Law
• Frequently Asked Questions

Ampere’s Swimming Rule

According to this rule, if a swimmer is imagined to swim along the conductor in the direction of the current, facing the needle, then the north pole of the needle is deflected to the left.

Magnetic Field

A stationary charge generates an electric field in its environment, and a moving charge generates a field in its environment that imposes a force on the moving charge. This field is referred to as the magnetic field, which is a vector quantity symbolised by the letter B. The nature of the magnetic field surrounding a conductor depends on the geometry of the conductor.

Motion of a Charged Particle in a Magnetic Field

Case 1: θ = 0⁰ 0r 180⁰

The path followed is a straight line.

As the angle between v and B is zero or 180°, a charged particle thrown parallel or antiparallel to a magnetic field will not experience any magnetic force. As a result, it will continue to travel in a straight line at a constant speed.

Case 2: θ = 90⁰

The path followed is circular.

A charged particle’s path is a circle when projected perpendicular to a uniform magnetic field. The magnetic Lorentz force works as a centripetal force, causing the charged particle to move at a constant speed in a circular path of radius R.

F = qvB = mv²/R

⇒ R = mv/qB

Angular velocity (ω) = v/R = qB/m

Time period of revolution, T = 2π/ω = 2πm/qB

Frequency of revolution = 1/T = qB/2πm

Biot-Savart Law

Consider the conductor’s infinitesimal element ds. The magnetic field dB produced by this element must be measured at a location P, that is, at a distance r away from it. Let r be the position vector and θ be the angle between them. The direction of ds is the same as the current direction.

The magnitude of the magnetic field dB is directly proportional to the current I and the element length ds, and it is inversely proportional to the square of the distance r, according to the Biot-Savart law. The direction of the magnetic field is perpendicular to the direction of the plane containing ds and r.

When the medium is a vacuum, the above expression stands true.

Here, μ₀ is the permeability of free space.

Ampere’s Circuital Law

According to Ampere’s circuital equation, the line integral of a steady magnetic field across a closed loop is equal to μ₀ times the total current (I_e) travelling through the surface bounded by the loop.

Application of Ampere’s Circuital Law

Magnetic Field Due to a Straight Infinite Current-Carrying Wire

Consider a current-carrying wire that is infinitely straight. We make an Amperian loop, which is a circle with a radius of r and a wire at its axis. As the magnetic field is tangential at all points on the loop, at all places, the magnetic field is of similar magnitude.

Now, applying Ampere’s Law,

B(2πr) = μ₀I

B = μ₀I/2πr

Magnetic Field due to a Solenoid

A solenoid is a cylindrical coil made of several turns of wire. The wire is coated with an insulating material so that even if the turns touch each other, they are electrically insulated.

Generally, the length of the solenoid is much greater than the radius of the transverse section of the solenoid. When the turns of the solenoid are closely spaced, each turn can be considered as a circular coil, and the net magnetic field at any point is the vector sum of the fields due to all the turns.

If the turns are closely wound, and the length is much greater than the radius, then the solenoid is treated as an ideal solenoid. For such a solenoid, the field outside is weaker than that inside the solenoid. Further, the field inside is uniform over a considerable volume and is along the axis of the solenoid. The figure below depicts a section of a long solenoid with an ‘n’ number of turns.

Using the Ampere circuital law, we can find the field. Consider a rectangular loop ABCD where the integration direction is anti-clockwise. For this loop,

μ₀I = B x a

If a solenoid has ‘n’ rounds of wire in a unit length, then the current is given by I = nai

Therefore, μ₀(nai) = B x a

⇒ B = μ₀(nai)/a

B = μ₀(ni)

Magnetic Force

When a charge is at rest, it is observed to experience almost no force. The charge q, on the other hand, is deflected towards the wire if it is given a velocity v in the direction of the current. As a result, we can conclude that a moving charged particle is acted upon by a magnetic field. Lorentz Force is the result of the interaction of electric and magnetic forces on a point charge.

At a given time, t, consider a point charge q travelling with velocity v and positioned at position vector r. If there is an electric field E and a magnetic field B at that location, the force on the electric charge q is equal to

This force is called the Lorentz force.

Magnetic Force on a Current-Carrying Conductor

Assume a current-carrying conductor is put in a uniform magnetic field, and l and A are the length and cross-sectional area, respectively. The free electrons in the wire have a drift velocity v_d that is opposite to the current direction. The force experienced by each electron

The total force experienced by the conductor is

The Force between Two Parallel Current-Carrying Wires

Currents i₁ and i₂ are carried in the same direction by two parallel wires. Each conductor creates a magnetic field at the other wire’s position. The other conductor is acted upon by this magnetic field. When current travels in the same direction across the conductors, they attract each other.

We’ll use the assumption that the conductors are infinitely long and calculate the force per unit length between them.

Let B₁ be the magnetic fields due to the first conductors.

Force on a sector of length ‘l’ for conductor 2 is

Moving Coil Galvanometer

In a uniform radial magnetic field, a galvanometer consists of a coil with many turns free to rotate around a fixed axis. The coil is wound around a cylindrical soft iron core, allowing the field to remain radial in all coil positions.

When the current flows through the coil, a torque acts on it. A torque is defined as force multiplied by the perpendicular distance between the forces.

τ = F x b

The torque acting on the coil’s single-loop ABCD.

τ = Bilb (Where l x b is the coil’s area A)

As a result, the torque acting on the n turns coil is given by

τ = niAB

The coil rotates as a result of the magnetic torque, and the phosphor bronze strip twists. The spring S linked to the coil, in turn, provides a counter torque or restoring torque kθ, resulting in a constant angular deflection.

Under equilibrium condition, kθ = niAB

Where k is the spring’s torsional constant, which is the restoring torque per unit twist. A pointer linked to the spring indicates the deflection on the scale.

θ = (nAB/k)i

The quantity nAB / k is a constant for a given galvanometer.

Galvanometer as Ammeter

The galvanometer cannot be used as an ammeter to measure the current in a circuit on its own.

A galvanometer is a highly sensitive instrument. For a current of the order of μA, it gives a full-scale deflection. Since the galvanometer has high resistance, it will modify the value of the current in the circuit.

To solve these challenges, a small resistance r_s, known as shunt resistance, is connected in parallel with the galvanometer coil, allowing the shunt to carry the majority of the current.

V_galvanometer = V_shunt

I_gR_g = (I – I_g)S

S = (I – I_g)/I_g

I_g = [S/(S + R_g)]I

I_g ∝ I

The deflection in the galvanometer is proportional to the current passing through it.

θ = (1/G)I_g

θ ∝ I_g

Therefore, θ ∝ I

The galvanometer is connected in parallel with the shunt resistance. As a result, the effective resistance, which is the resistance of an ammeter, may be calculated as

R_eff = R_GS/(R_G + S)

Galvanometer as Voltmeter

To use a galvanometer to determine the potential difference between two sections of a circuit, they must be connected in parallel to the circuit. Furthermore, it must take very little current; otherwise, the voltage measurement will cause a significant degree of disruption to the original setup. Normally, we try to minimise the disturbance caused by the measuring device to 1%. A large resistance R is connected in series with the galvanometer to ensure this.

G = Resistance of the galvanometer

R = Resistance connected in series

Let V volt represent the potential difference that the voltmeter will detect, and I_g represent the current. Therefore, the voltage between a and b is given by

V = I_gR + I_gG

= I_g (R + G)

⇒ R = (V/I_g) – G

This is the resistance that must be connected in series with the galvanometer in order to convert it into a voltmeter with a range of 0-V volts.

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