JEE Physics Notes: Moving Charges and Magnetism

Introduction to Moving Charges and Magnetism

Moving charges create magnetic fields, which is the basis for electromagnetism. The study of moving charges in a magnetic field is crucial for understanding electromagnets, electric motors, and generators.

Magnetic Field and Lorentz Force

A moving charge in a magnetic field experiences a force called the Lorentz force, given by:

F = q (E + v × B)

where:

• q = charge
• E = electric field
• v = velocity of the charge
• B = magnetic field

Motion of Charged Particles in a Magnetic Field

The trajectory of a charged particle depends on the direction of its velocity relative to the magnetic field.

Case 1: Velocity Perpendicular to Magnetic Field

When a charged particle moves perpendicular to the magnetic field, it follows a circular path with a radius given by:

r = (m v) / (q B)

The time period of revolution is:

T = (2πm) / (qB)

Case 2: Velocity at an Angle to Magnetic Field

The particle follows a helical path, with a pitch (distance moved along the field in one cycle) given by:

Pitch = vparallel × T

Biot-Savart Law

Biot-Savart Law describes the magnetic field due to a small current element:

dB = (μ₀ / 4π) * (I dl × r̂) / r²

where:

• μ₀ = permeability of free space
• I = current
• dl = small element of wire
• r = distance from the element

Ampère’s Circuital Law

Ampère’s Law states that the line integral of the magnetic field around a closed loop is proportional to the total current enclosed:

∮ B ⋅ dl = μ₀ Ienclosed

Magnetic Field due to Current-Carrying Conductors

Magnetic Field due to a Long Straight Wire

The magnetic field at a distance r from a long straight wire carrying current I is given by:

B = (μ₀ I) / (2πr)

Magnetic Field due to a Circular Current Loop

At the center of a circular loop of radius R, carrying current I, the field is:

B = (μ₀ I) / (2R)

Magnetic Field due to a Solenoid

A solenoid creates a uniform magnetic field inside it, given by:

B = μ₀ n I

where n is the number of turns per unit length.

Force Between Two Parallel Currents

Two parallel conductors carrying current exert a force on each other. The force per unit length is:

F/L = (μ₀ I₁ I₂) / (2πd)

If the currents flow in the same direction, the wires attract each other; if in opposite directions, they repel.

Torque on a Current-Carrying Loop

A current-carrying loop placed in a uniform magnetic field experiences a torque given by:

τ = NIAB si