Electro-Magnetism

Electro-Magnetism: Quick Revision Notes

1. Electric Charge and Field

• Charge Quantization: \( Q = \pm ne \), where \( n \) is an integer and \( e \) is the charge of an electron (\( 1.6 \times 10^{-19} \) C).
• Coulomb’s Law: The force between two point charges is \( F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \). It is conservative, central, and follows Newton’s 3rd Law.
• Electric Field (E): Force per unit charge. \( \vec{E} = \frac{\vec{F}}{q_0} \).
• Electric Field due to a Point Charge: \( E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \).
• Electric Dipole:
  • Moment: \( \vec{p} = q \times \vec{d} \) (direction from -q to +q).
  • Field on axial line: \( E = \frac{1}{4\pi\epsilon_0} \frac{2p}{r^3} \).
  • Field on equatorial line: \( E = \frac{1}{4\pi\epsilon_0} \frac{p}{r^3} \).
• Gauss’s Law: The net electric flux through a closed surface is \( \frac{1}{\epsilon_0} \) times the charge enclosed. \( \oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\epsilon_0} \).

2. Electric Potential and Capacitance

• Electric Potential (V): Work done to bring a unit positive charge from infinity to a point. \( V = \frac{1}{4\pi\epsilon_0} \frac{q}{r} \).
• Potential Difference: \( V_B – V_A = \frac{W_{A\to B}}{q_0} \).
• Relation between E and V: \( E = -\frac{dV}{dr} \). The electric field is in the direction of decreasing potential.
• Capacitance (C): \( C = \frac{Q}{V} \). Measured in Farad (F).
• Parallel Plate Capacitor: \( C = \frac{\epsilon_0 A}{d} \), where A is area and d is separation.
• Capacitors in Parallel: \( C_{eq} = C_1 + C_2 + C_3 + … \). Voltage same across all.
• Capacitors in Series: \( \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + … \). Charge same on all.
• Energy stored in Capacitor: \( U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C} \).

3. Current Electricity

• Electric Current (I): \( I = \frac{Q}{t} \). Rate of flow of charge. SI Unit: Ampere (A).
• Ohm’s Law: \( V = IR \), where R is resistance (Ohm, Ω).
• Resistance: \( R = \rho \frac{l}{A} \), where \( \rho \) is resistivity.
• Power: \( P = VI = I^2R = \frac{V^2}{R} \). SI Unit: Watt (W).
• Kirchhoff’s Laws:
  • Junction Rule (KCL): Sum of currents entering a junction equals sum leaving.
  • Loop Rule (KVL): Sum of potential differences around any closed loop is zero.

4. Magnetism

• Magnetic Field (B): Produced by moving charges/currents. SI Unit: Tesla (T).
• Biot-Savart Law: \( d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2} \).
• Ampere’s Circuital Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} \).
• Field due to a long straight wire: \( B = \frac{\mu_0 I}{2\pi r} \).
• Field at center of a circular loop: \( B = \frac{\mu_0 I}{2r} \) (for N turns, \( B = \frac{\mu_0 NI}{2r} \)).
• Force on a moving charge: \( \vec{F} = q (\vec{v} \times \vec{B}) \).
• Force on a current-carrying conductor: \( \vec{F} = I (\vec{l} \times \vec{B}) \).

5. Electromagnetic Induction & AC

• Faraday’s Law: The induced EMF is equal to the rate of change of magnetic flux. \( \epsilon = -\frac{d\phi_B}{dt} \).
• Lenz’s Law: The direction of the induced current opposes the change in flux that produced it. (The negative sign in Faraday’s law).
• Magnetic Flux: \( \phi_B = \vec{B} \cdot \vec{A} = BA\cos\theta \). SI Unit: Weber (Wb).
• Self Inductance (L): \( \epsilon = -L \frac{dI}{dt} \). SI Unit: Henry (H).
• Alternating Current (AC): \( I = I_0 \sin(\omega t) \) and \( V = V_0 \sin(\omega t) \), where \( I_0, V_0 \) are peak values.
• RMS Value: \( I_{rms} = \frac{I_0}{\sqrt{2}}, \quad V_{rms} = \frac{V_0}{\sqrt{2}} \).

6. Important Constants & Values

• Charge of electron (\( e \)): \( 1.6 \times 10^{-19} \) C
• Mass of electron (\( m_e \)): \( 9.1 \times 10^{-31} \) kg
• Permittivity of free space (\( \epsilon_0 \)): \( 8.85 \times 10^{-12} \) C²/N·m²
• Permeability of free space (\( \mu_0 \)): \( 4\pi \times 10^{-7} \) T·m/A
• Coulomb’s Constant (\( \frac{1}{4\pi\epsilon_0} \)): \( 9 \times 10^9 \) N·m²/C²
• Speed of light in vacuum (\( c \)): \( 3 \times 10^8 \) m/s
• 1 Tesla (T): = \( 10^4 \) Gauss (CGS unit)

7. Formula Sheet (At a Glance)

Coulomb's Force: \( F = k \frac{q_1 q_2}{r^2} \)
Electric Field (Point Charge): \( E = k \frac{q}{r^2} \)
Electric Potential: \( V = k \frac{q}{r} \)
Gauss's Law: \( \oint \vec{E} \cdot d\vec{A} = Q_{enc}/\epsilon_0 \)
Capacitance (Parallel Plate): \( C = \epsilon_0 A / d \)
Energy in Capacitor: \( U = \frac{1}{2}CV^2 \)
Ohm's Law: \( V = IR \)
Resistance: \( R = \rho l / A \)
Power: \( P = I^2R \)
Magnetic Field (Wire): \( B = \frac{\mu_0 I}{2\pi r} \)
Magnetic Force: \( F = qvB\sin\theta \)
Faraday's Law: \( \epsilon = -N \frac{d\phi_B}{dt} \)
AC RMS: \( V_{rms} = V_0 / \sqrt{2} \)