Moving Charges and Magnetism

MAGNETIC FORCE

4.2.1 Concept of Magnetic Field

**Magnetic field B(r)** is a vector field that exists in space around any current-carrying conductor or moving charge. Just as static charges produce an electric field E, **currents or moving charges produce a magnetic field B** in addition to an electric field.

Key properties:

Magnetic field is defined at each point in space

It obeys the **principle of superposition**: total magnetic field from multiple sources is vector addition of individual fields

Field can vary with time

In this chapter, we assume time-independent (static) fields

The magnetic field was discovered by **Hans Christian Oersted** in 1820 when he observed that a compass needle deflects near a current-carrying wire. The needle aligns **tangentially to circles concentric with the wire**, and the alignment reverses when current direction reverses.

---

4.2.2 LORENTZ FORCE

Definition and Formula

When a point charge q moves with velocity **v** in presence of both electric field **E** and magnetic field **B**, the total force experienced is called the **Lorentz force**:

**F = q[E(r) + v × B(r)]**

Or, **F = F_electric + F_magnetic** where F_magnetic = **q(v × B)**

The magnetic force component can be written as:

**F_B = q(v × B)** = **qvB sin θ n̂**

where:

q = charge of particle (in coulombs)

v = velocity of particle (in m/s)

B = magnetic field strength (in tesla)

θ = angle between velocity vector and magnetic field vector

n̂ = unit vector perpendicular to both v and B (given by right-hand rule)

Properties of Magnetic Force

**(i) Dependence on charge and velocity:**

Force is proportional to charge q

Force is proportional to velocity v

For negative charges, force is opposite in direction to positive charges

**Magnetic force is zero if particle is stationary** (v = 0) — unlike electric force which acts on stationary charges

**(ii) Direction of magnetic force:**

Given by **right-hand rule** (screw rule) for cross product v × B

Procedure: Point fingers in direction of v, curl them toward B, thumb points in direction of v × B

Force is **always perpendicular to both v and B**

Force is **perpendicular to velocity**, so it cannot do work on the particle

**(iii) Angle dependence:**

Force = 0 when v is parallel to B (sin 0° = 0)

Force = 0 when v is antiparallel to B (sin 180° = 0)

Force is **maximum when v ⊥ B** (sin 90° = 1)

**Physical meaning**: Magnetic force cannot change speed, only direction of motion

Unit of Magnetic Field

From **F = qvB**, the unit of B is derived as:

**[B] = F/(qv) = Newton/(Coulomb × m/s) = kg/(A·s²)**

**1 Tesla (T)** = Force on unit charge (1 C) moving perpendicular to B with speed 1 m/s

**1 Tesla = 1 Newton·second/(Coulomb·meter) = 1 kg/(A·s²)**

**Non-SI unit**: 1 Gauss = 10⁻⁴ Tesla

**Examples of magnetic field strengths**:

Earth's magnetic field ≈ 3.6 × 10⁻⁵ T

Typical bar magnet ≈ 0.01 T

Strong laboratory electromagnet ≈ 2-3 T

---

4.2.3 MAGNETIC FORCE ON CURRENT-CARRYING CONDUCTOR

Derivation from First Principles

Consider a straight conductor of:

Cross-sectional area = A

Length = l

Carrying current I

Charge carrier density = n

Charge per carrier = q

Drift velocity of carriers = v_d

Total number of charge carriers in conductor = **nAl**

Each carrier experiences magnetic force: **f = q(v_d × B)**

**Total force on all carriers:**

F = (nAl) × q(v_d × B)

Rearranging:

F = (nqv_d) × Al × B

Since **current I = nqv_d × A** (from Chapter 3 definition):

**nqv_d = I/A**

Therefore:

**F = (I/A) × A × l × B = I(l × B)**

Final Formula

**F = I(l × B)** or **|F| = IlB sin θ**

Where:

**l** = vector of magnitude l pointing in direction of current flow

I = current (scalar quantity in amperes)

B = magnetic field strength (tesla)

θ = angle between l and B

**Force is perpendicular to both l and B** (right-hand rule)

Key Points

Force is **maximum when l ⊥ B** (sin 90° = 1): **F_max = IlB**

Force is **zero when l || B** or **l antiparallel to B**

Force is **independent of the nature of charge carriers** — depends only on current I

For **curved or arbitrary-shaped wires**: **F = ∫I(dl × B)** (integral of all current elements)

Example 4.1: Suspended Wire in Magnetic Field

**Problem**: A straight wire of mass m = 200 g = 0.2 kg and length l = 1.5 m carries current I = 2 A. It is suspended horizontally in mid-air by a uniform horizontal magnetic field B. Find B.

**Solution**:

For mid-air suspension, upward magnetic force equals downward gravitational force:

**mg = IlB** (forces balance)

**B = mg/(Il)**

B = (0.2 kg × 9.8 m/s²)/(2 A × 1.5 m)

B = 1.96/3 = **0.65 T**

(Earth's field ≈ 4 × 10⁻⁵ T is negligible here)

Example 4.2: Direction of Lorentz Force

**Problem**: Magnetic field is along +y axis. Charged particle moves along +x axis. Find direction of Lorentz force for: (a) electron (b) proton.

**Solution**:

v × B: using right-hand rule, point fingers along +x (velocity), curl toward +y (field), thumb points along **+z axis**

**(a) For electron (negative charge):**

F = q(v × B) = (–e)(v × B) points along **–z axis** (opposite to v × B)

**(b) For proton (positive charge):**

F = q(v × B) = (+e)(v × B) points along **+z axis** (same as v × B)

---

4.3 MOTION IN A MAGNETIC FIELD

Circular Motion When v ⊥ B

When a charged particle moves perpendicular to a uniform magnetic field:

Magnetic force acts perpendicular to velocity

**Force does no work** (perpendicular to displacement)

**Kinetic energy is constant** (speed unchanged, only direction changes)

**Magnetic force acts as centripetal force**, causing circular motion

**Centripetal force equation**:

**qvB = mv²/r**

**Solving for radius:**

**r = mv/(qB)**

Key Features of Circular Motion

**Radius of circular path:**

**r = mv/(qB)**

Larger mass m → larger radius

Larger velocity v → larger radius

Larger charge q or field B → smaller radius

**Angular frequency (cyclotron frequency):**

From v = ωr:

**ω = v/r = qB/m**

Or, **ν = ω/2π = qB/(2πm)**

**Most important property**: **Frequency and period are INDEPENDENT of velocity or energy**

**Period of one revolution:**

**T = 2π/ω = 2πm/(qB)**

This independence of frequency from energy is the **basis of cyclotron operation**.

Helical Motion When v Has Component Along B

When velocity has components both parallel and perpendicular to B:

**v_perpendicular component**: produces circular motion in plane perpendicular to B

**v_parallel component**: continues unchanged (no force along field direction)

**Result**: Helical (spiral) motion around field lines

**Pitch of helix** (distance traveled along field per revolution):

**p = v_∥ × T = 2πmv_∥/(qB)**

---

4.4 MAGNETIC FIELD DUE TO CURRENT ELEMENT: BIOT-SAVART LAW

Statement of Biot-Savart Law

The magnetic field **dB** produced by an infinitesimal current element **I dl** at distance **r** is:

**dB = (μ₀/4π) × (I dl × r)/r³**

In scalar form:

**|dB| = (μ₀/4π) × (I dl sin θ)/r²**

Where:

**μ₀** = permeability of free space = **4π × 10⁻⁷ T·m/A**

**I** = current (amperes)

**dl** = current element vector (pointing in direction of current)

**r** = displacement vector from current element to field point P

**r** = magnitude of r

**θ** = angle between dl and r

**dB** = magnitude proportional to sin θ

Direction of Magnetic Field

**Direction of dB** is given by **right-hand rule** for cross product **dl × r**:

Point fingers along dl

Curl toward r

Thumb points in direction of dB

**dB is always perpendicular to plane containing dl and r**

Important Properties

**(i) Comparison with Coulomb's Law**:

| Feature | Coulomb's Law | Biot-Savart Law |

|---------|---------------|-----------------|

| Distance dependence | Inversely proportional to r² | Inversely proportional to r² |

| Source | Scalar (charge q) | Vector (current element I dl) |

| Field direction | Along displacement vector | Perpendicular to plane of dl and r |

| Angle dependence | None | Proportional to sin θ |

| Superposition | Applies | Applies |

**(ii) Angle dependence**:

Field = 0 along the direction of current (θ = 0, sin θ = 0)

Field = maximum perpendicular to current (θ = 90°, sin θ = 1)

**(iii) Total magnetic field** from a finite conductor:

**B = ∫dB = (μ₀I/4π) ∫(dl × r)/r³**

SI Units

**μ₀ = 4π × 10⁻⁷ T·m/A** (exact value, defined in SI)

This is the **permittivity-permeability relation**:

**c = 1/√(μ₀ε₀)** = 3 × 10⁸ m/s

where ε₀ = 8.85 × 10⁻¹² F/m

---

4.5 MAGNETIC FIELD ON THE AXIS OF A CIRCULAR CURRENT LOOP

Geometry and Setup

Consider:

**Circular loop** of radius **R** carrying steady current **I**

Loop in xy-plane centered at origin O

**Point P** on the axis (z-axis) at distance **x** from center

Calculate magnetic field at point P

Derivation Using Biot-Savart Law

**Step 1**: Consider infinitesimal element **dl** on the loop at angle φ

**Step 2**: From Biot-Savart law:

Distance from dl to point P: **r = √(R² + x²)**

Angle between dl and r: **sin θ = R/r = R/√(R² + x²)**

**Step 3**: Magnetic field from element dl:

**dB = (μ₀I/4π) × (R/√(R² + x²))² × dl/(R² + x²)**

**Step 4**: **Direction of dB**: Perpendicular to plane containing dl and r. Components:

**dB_parallel** to axis = dB sin(angle from radial) — **contributes to net field**

**dB_perpendicular** to axis — **cancels due to symmetry**

**Step 5**: Only **axial component** survives integration:

**dB_axial = (μ₀I/4π) × (R/(R² + x²)^(3/2)) × dl**

**Step 6**: Integrate around entire loop:

**B = ∫dB_axial = (μ₀I/4π) × (R/(R² + x²)^(3/2)) × ∫dl**

Since **∫dl = 2πR** (circumference):

**B = (μ₀I/4π) × (R/(R² + x²)^(3/2)) × 2πR**

Final Formula for Axial Field

**B = (μ₀IR²)/(2(R² + x²)^(3/2))**

Or equivalently:

**B = (μ₀I)/(2R) × (R²/(R² + x²)^(3/2))**

Where:

**B** = magnetic field at distance x from loop center on axis (tesla)

**I** = current in loop (amperes)

**R** = radius of loop (meters)

**x** = distance of point from loop center along axis (meters)

Special Cases

**(i) At center of loop (x = 0)**:

**B_center = μ₀I/(2R)**

This is the **maximum field** on the axis.

**(ii) Very far from loop (x >> R)**:

**B ≈ (μ₀I × 2πR²)/(4πx³) = (μ₀ × 2m)/(4πx³)**

where **m = IA = I × πR²** is the **magnetic dipole moment**

This shows field behaves like that of a **magnetic dipole** at large distances.

**(iii) Field direction**:

Along the **axis**, determined by **right-hand rule**

Curl fingers in direction of current, thumb points in direction of B

If current is counterclockwise when viewed from above, B points upward

Key Physics Points

**Magnetic field is NOT zero inside the loop** (unlike electric field in uniformly charged ring)

**Field is uniform-like near center**, but **non-uniform away from center**

**Field decreases as 1/x³** far from loop (dipole behavior)

**N-turn coil**: multiply result by N: **B = (μ₀NIR²)/(2(R² + x²)^(3/2))**

Example 4.3: Radius and Frequency of Electron in Magnetic Field

**Problem**: Electron (m = 9 × 10⁻³¹ kg, q = 1.6 × 10⁻¹⁹ C) moves at v = 3 × 10⁷ m/s perpendicular to B = 6 × 10⁻⁴ T. Find: (a) radius, (b) frequency, (c) energy in keV.

**Solution**:

**(a) Radius**:

**r = mv/(qB)**

r = (9 × 10⁻³¹ kg × 3 × 10⁷ m/s)/(1.6 × 10⁻¹⁹ C × 6 × 10⁻⁴ T)

r = (27 × 10⁻²⁴)/(9.6 × 10⁻²³) = 0.28 m = **28 cm**

**(b) Frequency**:

**ν = qB/(2πm)**

ν = (1.6 × 10⁻¹⁹ × 6 × 10⁻⁴)/(2π × 9 × 10⁻³¹)

ν = (9.6 × 10⁻²³)/(5.65 × 10⁻³⁰) = **1.7 × 10⁷ Hz = 17 MHz**

Alternatively: **ν = v/(2πr)** = (3 × 10⁷)/(2π × 0.28) ≈ 1.7 × 10⁷ Hz ✓

**(c) Energy**:

**E = ½mv²** = ½ × 9 × 10⁻³¹ × (3 × 10⁷)²

E = ½ × 9 × 10⁻³¹ × 9 × 10¹⁴ = **40.5 × 10⁻¹⁷ J = 4.05 × 10⁻¹⁶ J**

Converting to eV: **E = 4.05 × 10⁻¹⁶ / (1.6 × 10⁻¹⁹) = 2531 eV ≈ 2.5 keV**

Example 4.4: Magnetic Field from Current Element

**Problem**: Current element **Δl = 1 cm** (along +x axis) at origin carries **I = 10 A**. Find magnetic field at point P on y-axis at distance y = 0.5 m.

**Solution**:

Using **|dB| = (μ₀/4π) × (I|dl| sin θ)/r²**

Given:

|dl| = 0.01 m

I = 10 A

r = 0.5 m (distance from element to P)

θ = 90° (dl along x, r along y, so perpendicular)

μ₀/(4π) = 10⁻⁷ T·m/A

**|dB| = (10⁻⁷) × (10 × 0.01 × 1)/(0.5)²**

|dB| = (10⁻⁷ × 0.1)/0.25 = 10⁻⁸/0.25 = **4 × 10⁻⁸ T = 40 nT**

**Direction**: dl × r = (Δx î) × (y ĵ) = Δx·y (î × ĵ) = Δx·y k̂

So **B points in +z direction** (out of page)

---

EXAM-IMPORTANT POINTS AND COMMON MISCONCEPTIONS

Misconception 1: "Magnetic force does work"

**Fact**: Magnetic force is always perpendicular to velocity, so it does **zero work**. It changes direction, not speed.

Misconception 2: "Static charge experiences magnetic force"

**Fact**: Magnetic force **F = qv × B** requires v ≠ 0. Stationary charges experience NO magnetic force.

Misconception 3: "Lorentz force applies only to moving charges"

**Fact**: **Total force = qE + q(v × B)**. Electric force acts on all charges (moving or not); magnetic force only on moving charges.

Misconception 4: "Magnetic field is scalar like electric potential"

**Fact**: **B is a vector field** with direction and magnitude. Direction given by right-hand rule.

Misconception 5: "Cyclotron frequency depends on energy"

**Fact**: **ν = qB/(2πm) is independent of energy** — this is why cyclotron works!

---

QUICK FORMULA REFERENCE FOR BOARD EXAMS

| Formula | When to Use | Units |

|---------|-------------|-------|

| **F = q(v × B)** | Single moving charge | N = C·(m/s)·T |

| **F = IlB sin θ** | Current-carrying wire | N = A·m·T |

| **r = mv/(qB)** | Circular motion | m = kg·(m/s)/(C·T) |

| **ν = qB/(2πm)** | Cyclotron frequency | Hz = C·T/kg |

| **p = 2πmv_∥/(qB)** | Helix pitch | m |

| **dB = (μ₀/4π)(I dl sin θ)/r²** | Field from element | T |

| **B = μ₀IR²/[2(R²+x²)^(3/2)]** | Loop axial field | T |

| **B_center = μ₀I/(2R)** | At loop center | T |

---

PAST BOARD EXAM QUESTION PATTERNS

1. **Calculation type**: Find radius, frequency, or energy of charged particle in magnetic field → Use **r = mv/(qB)** and **ν = qB/(2πm)**

2. **Direction type**: Determine direction of force using right-hand rule → Draw diagram, apply cross product rule

3. **Biot-Savart application**: Calculate field due to current element or loop → Use formulas directly with given geometry

4. **Multiple concepts**: Combine motion equations with Lorentz force in circuits → Apply force balance and circular motion equations

5. **Qualitative**: Explain why needle deflects, why helix forms, why frequency is independent → Use physical principles