Unit 3: Magnetic Field

1. Magnetic Field (B) and Force

A Magnetic Field (B) is a vector field created by moving electric charges (currents). It is defined by the force it exerts on other moving charges.

Magnetic Force between Current Elements

The force on a moving charge q with velocity v is given by the Lorentz Force Law:

Lorentz Force: F = q(E + v × B)

In this unit, we focus on the magnetic part: F_mag = q(v × B).

Since a current is a stream of moving charges, a current-carrying wire also experiences a force in a B-field. The differential force (dF) on a small segment of wire (dl) carrying current (I) is:

Force on wire element: dF = I(dl × B)

2. Biot-Savart's Law and Applications

The Biot-Savart Law is the fundamental law of magnetostatics. It is used to calculate the magnetic field (dB) generated at a point by a small element of a current-carrying wire (Idl).

Biot-Savart Law: dB = (μ₀I / 4π) (dl × r̂) / r²

Where:

• μ₀ is the permeability of free space (4π × 10^-7 T·m/A).
• I is the current.
• dl is the vector length element of the wire.
• r is the distance from the element (dl) to the point.
• r̂ is the unit vector pointing from the element to the point.

To find the total B-field, one must integrate dB over the entire wire: B = ∫ dB.

Simple Applications of Biot-Savart's Law

1. B-field of a Long Straight Wire

By integrating the Biot-Savart law for a straight wire, the B-field at a perpendicular distance 'r' from an infinitely long wire is:

Formula: B = μ₀I / (2πr)

The field lines are concentric circles around the wire (direction given by the Right-Hand Grip Rule).

2. B-field of a Circular Loop (on its axis)

For a circular loop of radius 'R' carrying current 'I', the B-field at a point 'z' along the central axis is:

Formula: B(z) = (μ₀IR²) / (2(R² + z²)^3/2)

At the center of the loop (z=0):

Formula: B_center = μ₀I / (2R)

3. Helmholtz Coil

A Helmholtz coil consists of two identical circular coils (radius R, N turns) placed parallel to each other, separated by a distance d = R. They carry the same current I in the same direction.

Purpose: This specific arrangement creates a highly uniform magnetic field in the region between the coils. It is widely used in experiments (like measuring e/m of an electron) that require a known, uniform B-field.

3. Current Loop as a Magnetic Dipole

At large distances (z >> R), the B-field on the axis of a current loop becomes:

B(z) ≈ (μ₀IR²) / (2z³) = (μ₀ / 2π) (I(πR²)) / z³

This has the same 1/z³ dependence as the E-field of an electric dipole.

We define the magnetic dipole moment (μ) of the current loop:

Formula: μ = I * A

Where A is the vector area of the loop (Magnitude = Area, Direction = normal to the loop via Right-Hand Rule).

So, B(z) ≈ (μ₀ / 2π) (μ / z³). This shows the strong analogy between electric and magnetic dipoles.

4. Ampere's Circuital Law and Applications

Ampere's Law relates the line integral of the magnetic field B around a closed loop (an "Amperian loop") to the total current (I_enclosed) passing through that loop.

Ampere's Law: ∮ B ⋅ dl = μ₀I_enclosed

Applications of Ampere's Law

1. Ideal Solenoid

A solenoid is a long coil of wire. For an ideal solenoid (infinite length, tightly packed), we can find the field inside.

Using a rectangular Amperian loop, we find:

• Field Inside: B = μ₀nI (Uniform)
• Field Outside: B = 0

Where 'n' is the number of turns per unit length (n = N/L).

2. Toroid

A toroid is a solenoid bent into a donut shape. We use a circular Amperian loop of radius 'r' inside the coils.

B(2πr) = μ₀(NI), where N is the total number of turns.

Formula (inside toroid): B = (μ₀NI) / (2πr)

The field is not perfectly uniform (it depends on r), but it is contained entirely within the toroid (B=0 outside).

5. Properties of B (Divergence and Curl)

These are two of Maxwell's Equations, which describe the behavior of B in differential form.

Divergence of B

Formula: ∇ ⋅ B = 0

Physical Meaning: This is the mathematical statement that there are no magnetic monopoles. The divergence (∇⋅) measures the "outflow" from a point. Since ∇⋅B is always zero, there are no "sources" or "sinks" of magnetic field. Magnetic field lines always form closed loops.

Curl of B

Formula: ∇ × B = μ₀J

Physical Meaning: This is Ampere's Law in differential form. The curl (∇×) measures the "circulation" or "curliness" of a field. This equation states that circulating magnetic fields are produced by current densities (J).

6. Magnetic Force (Lorentz Force)

1. Force on a Point Charge

The magnetic part of the Lorentz force:

Formula: F = q(v × B)

Key Properties:

• The force F is always perpendicular to both v and B.
• Since F is perpendicular to v, the magnetic force does no work (W = ∫ F⋅dl = 0) on a free charge.
• It can change the *direction* of a particle's motion (causing it to circle), but not its *speed* or *kinetic energy*.

2. Force on a Current Carrying Wire

This is the macroscopic version of the force on point charges. The total force F on a wire is the integral of the differential force dF = I(dl × B).

For a straight wire of length L in a uniform field B:

Formula: F = I(L × B) (Magnitude: F = ILB sin(θ))

3. Force between Current Elements (e.g., Two Parallel Wires)

Wire 1 creates a field B₁ = μ₀I₁ / (2πd).

Wire 2 experiences a force from this field: F₂ = I₂(L × B₁).

The force per unit length (F/L) between them is:

Formula: F/L = μ₀I₁I₂ / (2πd)

• Parallel currents (same direction): Attract
• Anti-parallel currents (opposite direction): Repel

7. Torque on a Current Loop

When a current loop (like in a motor) is placed in a uniform magnetic field B, the forces on opposite sides create a twisting force, or torque (τ).

The torque tries to align the loop's magnetic dipole moment (μ) with the external field B.

Formula: τ = μ × B

The magnitude is τ = μB sin(θ), where θ is the angle between μ and B.

The potential energy (U) of the dipole in the field is:

Potential Energy: U = -μ ⋅ B = -μB cos(θ)

The energy is minimum (stable equilibrium) when θ=0 (μ aligned with B) and maximum (unstable) when θ=180° (μ anti-aligned).