Moving Charges and Magnetism Class 12 Notes

Key Terms and Definitions

Lorentz Force

The total force experienced by a charge 'q' moving with velocity 'v' in a region with both electric field 'E' and magnetic field 'B'. Formula: F = q[E + (v × B)].

Magnetic Field (B)

A vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. Its SI unit is the Tesla (T).

Biot-Savart Law

A law that gives the magnetic field produced by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current.

Ampere's Circuital Law

States that the line integral of the magnetic field B around any closed loop is equal to μ₀ times the total current I enclosed by the loop. Formula: ∮ B ⋅ dl = μ₀I_enc.

Solenoid

A long coil of wire wound in the form of a helix. When current flows through it, it produces a nearly uniform magnetic field in its interior.

Toroid

A hollow circular ring on which a large number of turns of a wire are closely wound. It can be viewed as a solenoid bent into a circle.

Cyclotron Frequency

The frequency of revolution of a charged particle in a cyclotron, which is independent of its energy or the radius of its orbit. Formula: ν = qB / (2πm).

Magnetic Dipole Moment (m)

A vector quantity that represents the strength and orientation of a magnet or a current loop. For a loop with N turns and current I, its magnitude is m = NIA, where A is the area of the loop.

Biot-Savart Law Explained

The Biot-Savart Law is fundamental to magnetostatics, analogous to Coulomb's Law in electrostatics. It allows us to calculate the magnetic field produced at a point due to a small segment of a current-carrying wire, known as a current element (Idl). The law states that the magnetic field dB at a position vector r from the current element Idl is directly proportional to the current I, the length of the element dl, and the sine of the angle (θ) between the direction of the current and the position vector. It is inversely proportional to the square of the distance r.

In vector form, the law is expressed as:
dB = (μ₀/4π) * (I dl × r̂) / r²

Where:

• dB is the differential magnetic field vector.
• μ₀ is the permeability of free space (4π × 10⁻⁷ T·m/A).
• I is the current flowing through the wire.
• dl is the differential length vector of the wire in the direction of the current.
• r̂ is the unit vector pointing from the current element towards the point of observation.
• r is the distance from the element to the point.

The direction of the magnetic field dB is perpendicular to the plane containing both dl and r, and can be found using the right-hand screw rule for the cross product. To find the total magnetic field B due to the entire wire, we must integrate this expression over the entire length of the wire.

Key Formulas and Geometries

• Lorentz Force: F = q(E + v × B). Magnetic force component is F_m = q(v × B). Magnitude: F_m = qvB sinθ.
• Force on a Current-Carrying Wire: F = I(L × B). Magnitude: F = ILB sinθ.
• Magnetic Field of a Straight Infinite Wire: B = (μ₀I) / (2πr).
• Magnetic Field at the Centre of a Circular Loop: B = (μ₀I) / (2R). For N turns, B = (μ₀NI) / (2R).
• Magnetic Field on the Axis of a Circular Loop: B = (μ₀IR²) / (2(x² + R²)^(3/2)).
• Ampere's Circuital Law: ∮ B ⋅ dl = μ₀I_enc.
• Magnetic Field inside a long Solenoid: B = μ₀nI, where n is the number of turns per unit length (N/L).
• Magnetic Field inside a Toroid: B = μ₀nI, where n is the number of turns per unit length (N/2πr). The field is zero outside the toroid.
• Torque on a Current Loop: τ = m × B = NI(A × B). Magnitude: τ = NIAB sinθ.
• Cyclotron Motion: Radius r = mv / (qB), Time period T = 2πm / (qB).

Force Between Two Parallel Currents

Working of a Moving Coil Galvanometer (MCG)

1. Step 1: Deflecting Torque —
2. Step 2: Restoring Torque —
3. Step 3: Equilibrium —
4. Step 4: Current Measurement —

Worked Example

• {"title":"Force on a Wire in a Magnetic Field","bodyMarkdown":"A straight wire of length 20 cm carries a current of 5 A. It is placed in a uniform magnetic field of 0.2 T, making an angle of 30° with the field. Calculate the magnetic force on the wire.\n\nSolution:\n Given: L = 20 cm = 0.2 m, I = 5 A, B = 0.2 T, θ = 30°.\n Formula: F = ILB sinθ\n Calculation: F = (5 A) (0.2 m) (0.2 T) sin(30°)\n F = (5 0.2 0.2) (0.5) = 0.2 0.5 = 0.1 N.\n Answer: The force on the wire is 0.1 N."}

Board Exam Traps and Tips

Direction, Direction, Direction! A majority of errors in this chapter come from incorrectly determining the direction of force or field.

• For Force: Use Fleming's Left-Hand Rule (for force on a conductor) or the vector cross product rule for F = q(v × B). Remember FBI: Force, Field, Current.
• For Field: Use the Right-Hand Thumb Rule. Thumb points in the direction of current (I), and curled fingers give the direction of the magnetic field (B).
• Vector Notation: Pay close attention to questions given in vector (i, j, k) format. Use the determinant method for cross products carefully. A sign error will change the entire direction.
• Solenoid vs. Toroid: Remember the magnetic field is zero outside a long solenoid and an ideal toroid. For a finite solenoid, the field near the ends is half the field at the center (B_end = μ₀nI / 2).

Practice Questions with Solutions

• Under what two conditions is the magnetic force on a moving charge in a uniform magnetic field equal to zero? 1. When the charge is stationary (v=0). 2. When the charge moves parallel or anti-parallel to the magnetic field (θ = 0° or 180°).
• How can a moving coil galvanometer be converted into an ammeter? By connecting a low-resistance wire, called a shunt (S), in parallel with the galvanometer coil.
• What is the primary function of a cyclotron? To accelerate charged particles (like protons, deuterons) to very high energies using both electric and magnetic fields.
• What is the SI unit of magnetic permeability (μ₀)? Tesla-meter per Ampere (T·m/A) or Henry per meter (H/m).

Frequently Asked Questions

Frequently Asked Questions