Moving Charges and Magnetism
Introduction
Electricity and magnetism are not separate phenomena but two interconnected aspects of a single interaction called electromagnetism. A stationary charge produces only an electric field because it exerts force only due to electrostatic interaction. However, when the same charge starts moving, it behaves like a source of magnetic effects as well.
This happens because motion of charge creates a disturbance in the surrounding electric field, which results in a magnetic field. This idea is extremely important because it explains how current-carrying wires, motors, and generators work in real life.
NOTE: Every magnetic effect in this chapter ultimately originates from moving charges. This is the core idea behind all formulas.
1. Magnetic Field
Definition: A magnetic field is a region around a moving charge or current-carrying conductor where another moving charge experiences a force.
A key point often misunderstood is that magnetic field does NOT act on stationary charges. This is why a charge must be moving to experience magnetic force.
Magnetic field lines are a visualization tool. They do not physically exist, but they help us understand direction and strength of field.
- Field lines emerge from North pole and enter South pole outside magnet
- Inside magnet, they move from South to North completing a loop
- Denser field lines indicate stronger magnetic field
The reason field lines never intersect is because magnetic field at any point has a unique direction. If they crossed, direction would become ambiguous, which is physically impossible.
Important: Magnetic field is a vector quantity and direction is given by tangent to field line.
2. Biot–Savart Law
Biot–Savart law is fundamental because it gives the magnetic field due to a small current element. It is similar to Coulomb’s law in electrostatics but for magnetism.
dB = (μ₀ / 4π) × (I dl sinθ / r²)
Each term has physical meaning:
- I: More current → more moving charges → stronger magnetic field
- dl: Small element of conductor contributes individually
- sinθ: Maximum effect when element is perpendicular
- 1/r²: Field decreases rapidly with distance
This law is essential in cases where symmetry is not simple. It allows calculation of field from any shaped conductor.
Reasoning: Always check symmetry. If symmetry is present → use Ampere’s Law. If not → use Biot–Savart Law.
Important Result
B = μ₀ I / 2R
At the center of a circular loop, every current element contributes equally because distance is same. All components add in same direction, giving maximum field.
3. Ampere’s Circuital Law
Ampere’s law is a shortcut version of Biot–Savart law used in symmetric cases. Instead of calculating every small element, we use a closed loop concept.
∮ B · dl = μ₀ Ienc
This law is based on symmetry. It works best when magnetic field is constant along a chosen loop.
(a) Straight Wire
Magnetic field forms concentric circles around a straight wire because moving charges create circular magnetic influence around the direction of current.
B = μ₀ I / 2πr
(b) Solenoid
Inside a solenoid, fields from each loop add up in same direction, producing strong uniform field. Outside, they cancel out.
B = μ₀ n I
(c) Toroid
In toroid, magnetic field is confined inside circular path because symmetry forces cancellation outside.
B = μ₀ N I / 2πr
4. Lorentz Force
Lorentz force explains how magnetic fields affect moving charges. It is responsible for circular and helical motion of particles.
F = q(v × B)
The cross product indicates that force is always perpendicular to both velocity and magnetic field, meaning magnetic force does no work.
F = qvB sinθ
If velocity is parallel to magnetic field, force becomes zero because no perpendicular component exists.
Motion Reasoning
r = mv / qB
Radius increases if mass or velocity increases, but decreases if charge or magnetic field increases. This explains particle accelerators.
T = 2πm / qB
Time period is independent of velocity because magnetic force only changes direction, not speed.
5. Force on Current Carrying Conductor
A conductor contains many moving charges. When placed in magnetic field, all charges experience force, resulting in net force on wire.
F = ILB sinθ
This principle is used in electric motors where current and magnetic field interaction produces rotational motion.
6. Torque on Current Loop
Torque arises because forces on opposite sides of loop form a couple, causing rotation.
τ = NIAB sinθ
M = NIA
Magnetic dipole moment determines how strongly a loop behaves like a magnet.
7. Moving Coil Galvanometer
It works because current in coil experiences torque in magnetic field. Greater current means larger deflection.
I = kθ / NBA
This relationship allows conversion into ammeter or voltmeter depending on resistance added.
8. Cyclotron
Cyclotron accelerates charged particles using repeated circular motion in magnetic field and alternating electric field.
f = qB / 2πm
Frequency remains constant because it depends only on charge, mass, and magnetic field.
Limitation arises at relativistic speeds because mass increases, breaking constant frequency condition.
9. Electric vs Magnetic Field
| Property |
Electric Field |
Magnetic Field |
| Source |
Stationary charges |
Moving charges |
| Work done |
Changes energy |
No work done |
| Effect |
Changes speed |
Changes direction only |
Conclusion
10. Important Numericals (With Detailed Solutions)
Q1. Magnetic field at center of circular loop
A loop of radius 0.1 m carries current 2 A. Find magnetic field at center.
Formula: B = μ₀ I / 2R
Substitute: μ₀ = 4π × 10⁻⁷, I = 2 A, R = 0.1 m
B = (4π × 10⁻⁷ × 2) / (2 × 0.1) = 2π × 10⁻⁶ T
Answer: 2π × 10⁻⁶ T
Reason: Field is directly proportional to current and inversely proportional to radius.
Q2. Force on moving charge
A charge 2 μC moves with velocity 3×10⁶ m/s perpendicular to B = 0.2 T. Find force.
F = qvB = (2×10⁻⁶)(3×10⁶)(0.2)
F = 1.2 N
Answer: 1.2 N
Reason: Maximum force occurs at 90° angle.
Q3. Radius of circular path
Electron enters B = 10⁻³ T with v = 10⁶ m/s.
r = mv / qB
m = 9×10⁻³¹ kg, q = 1.6×10⁻¹⁹ C
r ≈ 5.6 × 10⁻³ m
Answer: 5.6 mm
Reason: Light mass → small radius.
Q4. Force on conductor
Wire length 0.5 m, I = 4 A, B = 0.3 T, θ = 90°
F = ILB = 4 × 0.5 × 0.3 = 0.6 N
Q5. Magnetic field due to straight wire
I = 5 A, r = 2 cm
B = μ₀ I / 2πr
B = (4π×10⁻⁷ × 5) / (2π × 0.02)
B = 5 × 10⁻⁵ T
Q6. Solenoid field
n = 1000 turns/m, I = 2 A
B = μ₀ n I = 4π×10⁻⁷ × 1000 × 2
B = 8π × 10⁻⁴ T
Q7. Torque on loop
N = 10, I = 2 A, A = 0.05 m², B = 0.2 T, θ = 90°
τ = NIAB = 10 × 2 × 0.05 × 0.2
τ = 0.2 N·m
Q8. Time period in magnetic field
m = 10⁻³ kg, q = 2 C, B = 0.5 T
T = 2πm / qB
T = 2π × 10⁻³ / (2 × 0.5)
T = 6.28 × 10⁻³ s
Q9. Cyclotron frequency
q = 1.6×10⁻¹⁹, B = 0.1 T, m = 1.6×10⁻²⁷
f = qB / 2πm
f ≈ 1.6 × 10⁶ Hz
Q10. Magnetic dipole moment
N = 20, I = 3 A, A = 0.02 m²
M = NIA = 20 × 3 × 0.02 = 1.2 A·m²
Reason: Higher turns or current increases magnetic strength of loop.