Moving Charges and Magnetism in ONE SHOT - CUET Physics

Moving Charges and Magnetism

Magnetic Field and Its Origin

A magnetic field is the region around a magnet or magnetic material where magnetic forces can be experienced. The field influences materials like iron, cobalt, and nickel, and is created not only by magnets but also by moving electric charges. The concept was experimentally established by Oersted, who observed deflection in a magnetic compass needle when current flowed through a nearby wire.

• Magnetic field is directly proportional to current and inversely proportional to distance from the wire.
• Deflection angle $\theta$ increases with current and decreases with distance…

Moving Charges and Magnetism

Magnetic Field and Its Origin

A magnetic field is the region around a magnet or magnetic material where magnetic forces can be experienced. The field influences materials like iron, cobalt, and nickel, and is created not only by magnets but also by moving electric charges. The concept was experimentally established by Oersted, who observed deflection in a magnetic compass needle when current flowed through a nearby wire.

• Magnetic field is directly proportional to current and inversely proportional to distance from the wire.
• Deflection angle $\theta$ increases with current and decreases with distance from the wire.

Biot-Savart Law

The Biot-Savart Law quantifies the magnetic field produced by a small current element. It provides both magnitude and direction, using vector cross product rules.

• For a current element $idl$ at distance $r$ and angle $\theta$:$$dB = \frac{\mu_0}{4\pi} \frac{idl \sin\theta}{r^2}$$
• Vector form:$$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \vec{r}}{r^3}$$
• $\mu_0$ (permeability of free space) $= 4\pi \times 10^{-7}$ T·m/A.

Relationship Between $\mu_0$, $\epsilon_0$, and Speed of Light

The speed of light $c$ in vacuum is related to the permeability and permittivity of free space.

• $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$
• $\mu_0 = 4\pi \times 10^{-7}$ T·m/A
• $\epsilon_0 = 8.85 \times 10^{-12}$ F/m
• In a medium: $v = \frac{c}{\sqrt{\mu_r \epsilon_r}}$, where $\mu_r$ and $\epsilon_r$ are relative permeability and permittivity.

Magnetic Field Due to Current-Carrying Conductors

The magnetic field around a straight conductor is derived using Biot-Savart Law. For finite and infinite wires, the formulas differ.

• Finite wire at point $P$:$$B = \frac{\mu_0 I}{4\pi r} (\sin\theta_1 + \sin\theta_2)$$
• Infinite wire:$$B = \frac{\mu_0 I}{2\pi r}$$
• Direction is determined by the right-hand (Cobra) rule: thumb in current direction, fingers curl in field direction.

Magnetic Field Due to Loops and Coils

The field at the center of a square or circular loop is calculated by summing contributions from each segment.

• Square loop (side $2a$) at center:

  $$B_{total} = \frac{\mu_0 I 2}{\pi a}$$

• Circular loop (radius $R$) at center:

  $$B = \frac{\mu_0 I}{2R}$$

• For $n$ turns: $B = \frac{\mu_0 n I}{2R}$

• Direction: use Cobra rule; for clockwise current, field is downward at center.

• On axis of circular loop (distance $x$ from center, radius $a$):

  $$B = \frac{\mu_0 I a^2}{2(x^2 + a^2)^{3/2}}$$

Ampere’s Circuital Law

Ampere’s Law relates the line integral of magnetic field around a closed loop to the total current enclosed.

• Statement: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}$
• For a straight wire, using a circular loop of radius $r$:$$B \cdot 2\pi r = \mu_0 I \implies B = \frac{\mu_0 I}{2\pi r}$$
• Only the enclosed current is considered; field is due to all currents.

Magnetic Field in a Solenoid

A solenoid is a coil with many turns, producing a uniform magnetic field inside.

• For a solenoid of length $l$, number of turns $N$, and current $I$:$$B = \mu_0 n I$$where $n = \frac{N}{l}$ is turn density.
• Field inside is straight and uniform; outside is negligible.

Force on Moving Charge in Magnetic Field (Lorentz Force)

A moving charge in a magnetic field experiences a force perpendicular to both velocity and field.

• Magnitude: $F = q v B \sin\theta$
• Direction: given by right-hand rule (Palm/Cobra rule).
• For $\theta = 90^\circ$, $F_{max} = q v B$
• Lorentz force (total): $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

Motion of Charged Particle in Uniform Magnetic Field

A charged particle entering a uniform magnetic field moves in a circular or helical path due to the perpendicular force.

• Radius of circular path:$$r = \frac{m v}{q B}$$
• Time period:$$T = \frac{2\pi m}{q B}$$
• Frequency:$$f = \frac{q B}{2\pi m}$$
• Angular frequency:$$\omega = \frac{q B}{m}$$
• For entry at angle $\theta$, radius uses $v \sin\theta$; pitch of helix is $v \cos\theta \times T$.

Force on Current-Carrying Conductor in Magnetic Field

A conductor of length $l$ carrying current $I$ in a magnetic field $B$ experiences a force.

• Magnitude: $F = B I l \sin\theta$
• Vector form: $\vec{F} = I (d\vec{l} \times \vec{B})$
• Direction: right-hand rule; force is perpendicular to both current and field.

Force Between Two Parallel Current-Carrying Wires

Two parallel wires carrying currents $I_1$ and $I_2$ at distance $r$ exert force on each other.

• Force per unit length:$$F = \frac{\mu_0 I_1 I_2}{2\pi r}$$
• Attractive if currents are in same direction, repulsive if opposite.
• Definition of 1 ampere: when two wires 1 m apart carry 1 A each, force per unit length is $2 \times 10^{-7}$ N/m.

Torque on a Current Loop in Magnetic Field and Moving Coil Galvanometer

A current loop in a magnetic field experiences torque, causing rotation. This principle is used in moving coil galvanometers.

• Torque on loop:

  $$\tau = n I A B \sin\theta$$

  where $n$ = number of turns, $A$ = area, $B$ = field, $\theta$ = angle between normal and field.

• Maximum torque: $\tau_{max} = n I A B$ (when $\theta = 90^\circ$)

• Restoring torque: $\tau_{restoring} = k \theta$, $k$ = torsional constant.

• At equilibrium: $n I A B = k \theta$

• Galvanometer constants:

  • Galvanometer constant: $I/\theta = k/(nAB)$
  • Current sensitivity: $\theta/I = nAB/k$
  • Voltage sensitivity: $nAB/(kR)$, where $R$ is resistance.

Conversion of Galvanometer to Ammeter and Voltmeter

A galvanometer can be converted to an ammeter by connecting a low resistance (shunt) in parallel, or to a voltmeter by connecting a high resistance in series.

• Ammeter conversion:

  • Shunt resistance: $R_s = \frac{I_G R_G}{I - I_G}$
  • $I$ = total current, $I_G$ = galvanometer current, $R_G$ = galvanometer resistance.

• Voltmeter conversion:

  • Series resistance: $R = \frac{V}{I_G} - R_G$
  • $V$ = desired voltage range, $I_G$ = full-scale deflection current.

• In both cases, the majority of current or voltage is diverted through the added resistance, allowing the galvanometer to measure larger values.

Exam-Focused Takeaways and Key Formulas

• Biot-Savart Law: $dB = \frac{\mu_0}{4\pi} \frac{idl \sin\theta}{r^2}$
• Infinite wire field: $B = \frac{\mu_0 I}{2\pi r}$
• Circular loop center field: $B = \frac{\mu_0 I}{2R}$
• Ampere’s Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}$
• Solenoid field: $B = \mu_0 n I$
• Lorentz force: $F = q v B \sin\theta$
• Radius of path: $r = \frac{m v}{q B}$
• Force on wire: $F = B I l \sin\theta$
• Force between wires: $F = \frac{\mu_0 I_1 I_2}{2\pi r}$
• Torque on loop: $\tau = n I A B \sin\theta$
• Galvanometer conversions: $R_s = \frac{I_G R_G}{I - I_G}$, $R = \frac{V}{I_G} - R_G$