CBSE Class 12 Physics Moving Charges and Magnetism Notes Set 03

1. Magnetic Field and Oersted's Experiment

In 1820 Hans Christian Oersted noticed that a magnetic compass needle placed near a wire is deflected when a current flows through the wire. Reversing the current reverses the deflection, and the deflection grows with the current. This showed that moving charges (currents) produce a magnetic field around them.
1. A stationary charge produces only an electric field. A moving charge produces both an electric field and a magnetic field.
2. The magnetic field B is a vector field. Its SI unit is the tesla (T); a smaller unit is the gauss (1 G = 10⁻⁴ T). The Earth's field is about 3.6 × 10⁻⁵ T.
3. A field coming out of the page is shown by dots (⊙) and a field going into the page by crosses (×).

2. Magnetic Force on a Moving Charge (Lorentz Force)

A charge q moving with velocity v in a magnetic field B experiences a force
\( \vec{F} = q(\vec{v} \times \vec{B}) \), magnitude F = qvB sin θ, where θ is the angle between v and B.
If an electric field E is also present, the total force (the Lorentz force) is \( \vec{F} = q[\vec{E} + \vec{v} \times \vec{B}] \).

Key features of the magnetic force:
1. It is zero for a charge at rest (v = 0) and for a charge moving parallel or antiparallel to B (θ = 0° or 180°).
2. It is maximum (qvB) when v is perpendicular to B.
3. It is always perpendicular to both v and B, so it does no work on the charge. It changes the direction of motion but not the speed or kinetic energy.
4. The force on a negative charge is opposite to that on a positive charge moving the same way.
Definition of tesla: B = 1 T when a charge of 1 C moving at 1 m s⁻¹ perpendicular to the field experiences a force of 1 N. So 1 T = 1 N s C⁻¹ m⁻¹ = 1 N A⁻¹ m⁻¹.

3. Motion of a Charged Particle in a Uniform Magnetic Field

(i) v perpendicular to B: The magnetic force provides the centripetal force, so the particle moves in a circle.
qvB = \( \frac{mv^2}{r} \)  ⇒  r = \( \frac{mv}{qB} \) = \( \frac{p}{qB} \)
Angular frequency: ω = \( \frac{qB}{m} \); frequency: ν = \( \frac{qB}{2\pi m} \); time period: T = \( \frac{2\pi m}{qB} \)
The period and frequency do not depend on the speed or the radius. A faster particle moves in a bigger circle but takes the same time per revolution.

(ii) v at an angle θ to B: Split v into v_∥ = v cos θ (along B) and v_⊥ = v sin θ (perpendicular to B). The parallel part is unaffected and the perpendicular part gives circular motion, so the path is a helix.
Radius r = \( \frac{m v_{\perp}}{qB} \); pitch (distance moved along B in one revolution) p = v_∥ T = \( \frac{2\pi m v_{\parallel}}{qB} \)

(iii) v parallel to B: No force acts, and the particle moves in a straight line with constant velocity.

4. Magnetic Force on a Current-Carrying Conductor

A straight conductor of length l carrying current I in a uniform field B experiences the sum of the forces on its moving charge carriers:
\( \vec{F} = I(\vec{l} \times \vec{B}) \), magnitude F = IlB sin θ, where \( \vec{l} \) points along the current.
1. F is maximum (IlB) when the conductor is perpendicular to B and zero when it is parallel to B.
2. The direction is given by the right-hand rule for the cross product (or Fleming's left-hand rule).
3. Force per unit length on a wire perpendicular to B: \( \frac{F}{l} \) = IB.

5. Biot–Savart Law

The magnetic field dB at a point P due to a small current element I dl is
\( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \vec{r}}{r^3} \), magnitude dB = \( \frac{\mu_0}{4\pi} \frac{I \, dl \sin\theta}{r^2} \)
where r is the distance from the element to P and θ is the angle between dl and r.
μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space, so \( \frac{\mu_0}{4\pi} \) = 10⁻⁷ T m A⁻¹.

Comparison with Coulomb's law:
1. Both fields fall off as 1/r² and both obey superposition.
2. The electric field is produced by a scalar source (charge); the magnetic field is produced by a vector source (I dl).
3. The electric field is along the line joining source and point; the magnetic field is perpendicular to the plane containing dl and r.
4. dB depends on the angle θ; it is zero at points along the line of the current element.
5. \( \frac{1}{\sqrt{\mu_0 \varepsilon_0}} \) = c = 3 × 10⁸ m s⁻¹, the speed of light in vacuum.

6. Magnetic Field on the Axis of a Circular Current Loop

For a circular loop of radius R carrying current I, the field at a point on its axis at distance x from the centre is
B = \( \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}} \), directed along the axis.
1. At the centre (x = 0): B = \( \frac{\mu_0 I}{2R} \). For a coil of N closely wound turns, B = \( \frac{\mu_0 N I}{2R} \).
2. Far from the loop (x ≫ R): B ≈ \( \frac{\mu_0 I R^2}{2x^3} \), which falls as 1/x³.
3. Right-hand thumb rule: Curl the fingers of the right hand in the direction of the current round the loop; the thumb points along the field at the centre.
4. Seen from one face, an anticlockwise current makes that face behave like a north pole; a clockwise current makes it a south pole.

7. Ampere's Circuital Law

The line integral of B around any closed loop is μ₀ times the total current threading the loop:
\( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enclosed}} \)
The closed loop (an Amperian loop) can have any shape. Currents outside it give zero net contribution to the integral. The sign of each current is fixed by the right-hand rule: with fingers curled along the direction of the loop, currents along the thumb count as positive. Like Gauss's law, it is most useful when symmetry makes B constant along the chosen loop.

(i) Infinitely long straight wire: Take a circle of radius r centred on the wire. B is tangential and has the same magnitude everywhere on it, so
B × 2πr = μ₀I  ⇒  B = \( \frac{\mu_0 I}{2\pi r} \)
The field lines are concentric circles around the wire, and B is inversely proportional to r. The direction follows the right-hand thumb rule: thumb along the current, curled fingers along B.

(ii) Long straight solenoid: A solenoid is a long wire wound as a close-packed helix. With n turns per unit length carrying current I, a rectangular Amperian loop with one side of length h inside gives Bh = μ₀(nh)I, so
B = μ₀nI (inside, far from the ends)
1. The field inside is uniform and directed along the axis; outside a long solenoid it is nearly zero.
2. At either end of a long solenoid, the field on the axis is about half the value at the centre: B = \( \frac{1}{2} \)μ₀nI.
3. Inserting a soft iron core greatly increases the field.

8. Force between Two Parallel Currents and the Ampere

Each wire sits in the field of the other. Wire a produces B_a = \( \frac{\mu_0 I_1}{2\pi d} \) at wire b, so the force per unit length on b is
f = \( \frac{F}{L} \) = \( \frac{\mu_0 I_1 I_2}{2\pi d} \)
1. Currents in the same direction attract; currents in opposite directions repel. (This is the reverse of charges, where likes repel.)
2. The forces on the two wires are equal and opposite, as Newton's third law requires.
Definition of the ampere: One ampere is the steady current which, flowing in each of two very long, straight, parallel conductors of negligible cross-section placed 1 m apart in vacuum, produces a force of 2 × 10⁻⁷ N per metre of length between them.

9. Torque on a Current Loop in a Uniform Magnetic Field

A rectangular coil of N turns, area A, carrying current I, placed in a uniform field B:
1. The net force on the loop is zero, because the forces on opposite sides are equal and opposite.
2. The forces on the two sides perpendicular to B form a couple, giving a torque
τ = NIAB sin θ, or in vector form \( \vec{\tau} = \vec{m} \times \vec{B} \)
where θ is the angle between the normal to the coil (the direction of m) and B.
3. τ is maximum (NIAB) when the plane of the coil is parallel to B (θ = 90°).
4. τ = 0 when the plane of the coil is perpendicular to B (θ = 0°, stable equilibrium; θ = 180°, unstable equilibrium).
5. The result holds for a plane loop of any shape, with A its area.

10. Current Loop as a Magnetic Dipole

A current loop behaves like a bar magnet. Its magnetic dipole moment is
\( \vec{m} = N I \vec{A} \), magnitude m = NIA, SI unit A m² (also J T⁻¹).
The direction of m is along the normal to the loop, given by the right-hand thumb rule (fingers along the current, thumb along m).
On the axis far from the loop (x ≫ R): B = \( \frac{\mu_0}{4\pi} \frac{2m}{x^3} \)

Analogy with the electric dipole:

The two differ in one way: an isolated magnetic pole has never been found, whereas isolated electric charges exist.

11. Moving Coil Galvanometer

Principle: A current-carrying coil placed in a magnetic field experiences a torque.
Construction: A rectangular coil of many turns is wound on a light frame and pivoted between the concave pole pieces of a strong permanent magnet. A soft iron cylinder sits inside the coil. Together these make the field radial, so the plane of the coil is always parallel to B and sin θ = 1 at every position. A spring provides the restoring torque, and a pointer moves over a scale.
Working: In equilibrium the magnetic torque balances the restoring torque of the spring:
NIAB = kφ  ⇒  φ = \( \frac{NAB}{k} \) I
where k is the torsional constant of the spring and φ the deflection. Since φ is proportional to I, the scale is linear.

Current sensitivity (deflection per unit current): \( \frac{\varphi}{I} \) = \( \frac{NAB}{k} \)
Voltage sensitivity (deflection per unit voltage): \( \frac{\varphi}{V} \) = \( \frac{NAB}{kR_G} \), where R_G (or G) is the resistance of the coil.
1. Current sensitivity can be raised by increasing N, A or B, or by using a spring with a smaller k.
2. Doubling the number of turns doubles the current sensitivity, but it also roughly doubles the coil resistance, so the voltage sensitivity stays nearly the same.

12. Conversion of a Galvanometer into an Ammeter and a Voltmeter

Let G be the galvanometer resistance and I_g the current for full-scale deflection.

(i) Ammeter: A small resistance S, called a shunt, is connected in parallel with the galvanometer so that most of the current bypasses the coil. For a full-scale reading of I,
I_gG = (I − I_g)S  ⇒  S = \( \frac{I_g G}{I - I_g} \)
Resistance of the ammeter = \( \frac{GS}{G + S} \), which is very small. An ideal ammeter has zero resistance; it is connected in series in the circuit.

(ii) Voltmeter: A large resistance R is connected in series with the galvanometer. For a full-scale reading of V,
V = I_g(G + R)  ⇒  R = \( \frac{V}{I_g} \) − G
Resistance of the voltmeter = G + R, which is very large. An ideal voltmeter has infinite resistance; it is connected in parallel across the element.

13. Quick Formula Recap