Electricity

Introduction to Electricity

1. Definition:

   • Electricity is the flow of electric charge, typically carried by electrons in a conductor.
   • It manifests in the form of electric current, voltage (potential difference), and resistance.

2. Basic Units:

   • Electric Charge ($Q$): The fundamental property of matter that gives rise to electric force.
     • SI Unit: Coulomb (C).
   • Electric Current ($I$): The rate of flow of electric charge.
     • Formula: $I = \frac{Q}{t}$
     • SI Unit: Ampere (A), where $1 \, A = 1 \, C/s$.
   • Voltage (Potential Difference): The energy per unit charge required to move a charge between two points.
     • SI Unit: Volt (V), where $1 \, V = 1 \, J/C$.
   • Resistance ($R$): The opposition to the flow of current.
     • SI Unit: Ohm ($\Omega$), where $1 \, \Omega = 1 \, V/A$.

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Ohm’s Law

1. Statement:

   • The current through a conductor is directly proportional to the voltage across it and inversely proportional to the resistance.
   • Formula: $V = IR$
     • $V$: Voltage.
     • $I$: Current.
     • $R$: Resistance.

2. Application:

   • Ohm’s law is applicable to conductors at constant temperature and materials that follow a linear relationship between voltage and current.

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Resistivity and Conductivity

1. Resistivity ($\rho$):

   • The property of a material that determines its resistance.
   • Formula: $R = \rho \frac{L}{A}$
     • $L$: Length of the conductor.
     • $A$: Cross-sectional area of the conductor.
     • $\rho$: Resistivity, which is a constant for a given material.

2. Conductivity ($\sigma$):

   • The ability of a material to conduct electric current.
   • Formula: $\sigma = \frac{1}{\rho}$

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Series and Parallel Circuits

1. Series Circuits:

   • Components are connected end-to-end.
   • Total Resistance: $R_{\text{total}} = R_1 + R_2 + \dots + R_n$
   • Current:
     • Same current flows through all components.
   • Voltage: $V_{\text{total}} = V_1 + V_2 + \dots + V_n$

2. Parallel Circuits:

   • Components are connected across the same two points.
   • Total Resistance: $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$
   • Current: $I_{\text{total}} = I_1 + I_2 + \dots + I_n$
   • Voltage:
     • Same voltage across all components.

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Electric Power

1. Definition:

   • The rate at which electrical energy is consumed or converted into another form (heat, light, etc.).
   • Formula: $P = VI = I^2 R = \frac{V^2}{R}$

2. SI Unit:

   • Watt (W), where $1 \, W = 1 \, J/s$.

3. Energy Consumption:

   • Energy used is the power consumed over time: $E = P \cdot t$

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Capacitors

1. Definition:

   • A capacitor is a device used to store electric charge and energy in an electric field.

2. Capacitance ($C$):

   • The ability of a capacitor to store charge per unit voltage.
   • Formula: $C = \frac{Q}{V}$
   • SI Unit: Farad (F), where $1 \, F = 1 \, C/V$.

3. Energy Stored in a Capacitor:

   • Formula: $E = \frac{1}{2} C V^2$

4. Capacitors in Series and Parallel:

   • Series: $\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$
   • Parallel: $C_{\text{total}} = C_1 + C_2 + \dots + C_n$

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Numerical Example

1. Example 1: A 10 $\Omega$ resistor is connected in series with a 20 $\Omega$ resistor. Find the total resistance.

   • Formula: $R_{\text{total}} = R_1 + R_2$
   • Substituting values: $R_{\text{total}} = 10 + 20 = 30 \, \Omega$

2. Example 2: A battery provides 12 V to a circuit with a resistance of 6 $\Omega$. Calculate the current.

   • Using Ohm's Law: $I = \frac{V}{R} = \frac{12}{6} = 2 \, A$

3. Example 3: A capacitor of capacitance 5 $\mu F$ is charged to 10 V. Find the energy stored in the capacitor.

   • Formula: $E = \frac{1}{2} C V^2$
   • Substituting values: $E = \frac{1}{2} \cdot 5 \times 10^{-6} \cdot 10^2 = 0.00025 \, J$

Electric Field and Electric Potential

1. Electric Field ($E$):

   • The region around a charged object where other charges experience a force.
   • Formula: $E = \frac{F}{q}$
     • $F$: Force on a test charge $q$.
   • For a point charge: $E = \frac{kQ}{r^2}$
     • $k$: Coulomb’s constant ($8.99 \times 10^9 \, N \cdot m^2 / C^2$).
     • $Q$: Source charge.
     • $r$: Distance from the charge.

2. Electric Potential ($V$):

   • The potential energy per unit charge at a point in an electric field.

   • Formula: $V = \frac{U}{q}$

     • $U$: Potential energy, $q$: Test charge.

   • For a point charge: $V = \frac{kQ}{r}$

3. Relation Between Electric Field and Potential:

   • Electric field is the negative gradient of electric potential: $E = -\frac{dV}{dr}$

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Coulomb’s Law

1. Statement:

   • Coulomb’s law describes the force between two point charges.
   • Formula: $F = k \frac{|Q_1 Q_2|}{r^2}$
     • $Q_1$ and $Q_2$: Magnitudes of the charges.
     • $r$: Distance between the charges.

2. Nature of Force:

   • Attractive force if charges are of opposite sign.
   • Repulsive force if charges are of the same sign.

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Gauss’s Law

1. Statement:

   • The total electric flux through a closed surface is proportional to the charge enclosed within that surface.
   • Formula: $\Phi_E = \oint E \cdot dA = \frac{Q_{\text{enc}}}{\epsilon_0}$
     • $\Phi_E$: Electric flux.
     • $E$: Electric field.
     • $dA$: Differential area.
     • $Q_{\text{enc}}$: Enclosed charge.
     • $\epsilon_0$: Permittivity of free space ($8.85 \times 10^{-12} \, C^2/N \cdot m^2$).

2. Applications:

   • Used to calculate electric fields of symmetrical charge distributions, such as spheres and cylinders.

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Capacitance in Parallel Plate Capacitors

1. Capacitance:

   • The ability of a capacitor to store charge.
   • Formula: $C = \frac{\epsilon_0 A}{d}$
     • $A$: Area of the plates.
     • $d$: Distance between the plates.

2. Dielectrics:

   • Materials that increase the capacitance of a capacitor by reducing the electric field between the plates.
   • Capacitance with a Dielectric: $C = \kappa \frac{\epsilon_0 A}{d}$
     • $\kappa$: Dielectric constant of the material.

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Current and Voltage

1. Electric Current ($I$):

   • The rate of flow of charge through a conductor.
   • Formula: $I = \frac{Q}{t}$

2. Ohm’s Law:

   • Relationship between current, voltage, and resistance.
   • Formula: $V = IR$

3. Electrical Power:

   • The rate at which electrical energy is consumed or converted to other forms (heat, light, etc.).
   • Formula: $P = I^2 R = \frac{V^2}{R}$

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Kirchhoff’s Laws

1. Kirchhoff’s Current Law (KCL):

   • The total current entering a junction equals the total current leaving the junction.
   • Formula: $\sum I_{\text{in}} = \sum I_{\text{out}}$

2. Kirchhoff’s Voltage Law (KVL):

   • The sum of the voltages around any closed loop in a circuit is zero.
   • Formula: $\sum V = 0$

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Numerical Examples

1. Example 1: Two charges of $+3 \, \mu C$ and $-2 \, \mu C$ are separated by $0.5 \, m$. Find the force between them.

   • Formula: $F = k \frac{|Q_1 Q_2|}{r^2}$
   • Substituting values: $F = 8.99 \times 10^9 \cdot \frac{|3 \times 10^{-6} \times (-2 \times 10^{-6})|}{(0.5)^2}$ $F = 8.99 \times 10^9 \cdot \frac{6 \times 10^{-12}}{0.25} = 2.16 \, N$

2. Example 2: A parallel plate capacitor has a plate area of $1 \, m^2$ and a plate separation of $1 \, mm$. Find its capacitance. (Use $\epsilon_0 = 8.85 \times 10^{-12} \, C^2/N \cdot m^2$)

   • Formula: $C = \frac{\epsilon_0 A}{d}$
   • Substituting values: $C = \frac{8.85 \times 10^{-12} \cdot 1}{1 \times 10^{-3}} = 8.85 \times 10^{-9} \, F = 8.85 \, nF$

Magnetic Fields

1. Magnetic Field ($B$):

   • A region in space where a magnetic force is experienced by a moving charge.
   • SI Unit: Tesla (T), where $1 \, T = 1 \, N \cdot s / C \cdot m$.
   • Magnetic fields are produced by moving charges (currents) or magnetic materials.

2. Magnetic Force on a Moving Charge:

   • Formula: $F = qvB \sin \theta$
     • $q$: Charge.
     • $v$: Velocity of the charge.
     • $B$: Magnetic field strength.
     • $\theta$: Angle between velocity and magnetic field.

3. Right-Hand Rule:

   • For the direction of magnetic force on a positive charge: Point the thumb in the direction of velocity, the fingers in the direction of the magnetic field, and the palm shows the direction of force.

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Ampere’s Law

1. Statement:

   • The magnetic field created by a current is proportional to the current and the path taken by the current.
   • Formula: $\oint B \cdot dl = \mu_0 I_{\text{enc}}$
     • $\mu_0$: Permeability of free space ($4\pi \times 10^{-7} \, T \cdot m/A$).
     • $I_{\text{enc}}$: Enclosed current.

2. Magnetic Field Due to a Long Straight Current-Carrying Wire:

   • Formula: $B = \frac{\mu_0 I}{2 \pi r}$
     • $r$: Distance from the wire.

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Faraday’s Law of Electromagnetic Induction

1. Statement:

   • A change in magnetic flux through a circuit induces an electromotive force (EMF) in the circuit.
   • Formula: $\mathcal{E} = -\frac{d\Phi_B}{dt}$
     • $\mathcal{E}$: Induced EMF.
     • $\Phi_B$: Magnetic flux.

2. Lenz’s Law:

   • The direction of the induced current is such that it opposes the change in magnetic flux.

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Self-Induction and Mutual Induction

1. Self-Induction:

   • The phenomenon where a changing current in a coil induces an EMF in the same coil.
   • Formula: $\mathcal{E} = -L \frac{dI}{dt}$
     • $L$: Self-inductance.

2. Mutual Induction:

   • When a changing current in one coil induces an EMF in a second coil.
   • Formula: $\mathcal{E}_2 = -M \frac{dI_1}{dt}$
     • $M$: Mutual inductance.

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Electromagnetic Waves

1. Definition:

   • Electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields, propagating through space.
   • They do not require a medium and can travel through a vacuum.
   • The speed of electromagnetic waves in a vacuum is the speed of light: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

2. Properties:

   • They exhibit wave-particle duality and can be described by the same wave equations as light.
   • Electromagnetic waves include radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays, and gamma rays.

3. Relation Between Electric and Magnetic Fields:

   • In an electromagnetic wave, the electric and magnetic fields are perpendicular to each other and to the direction of wave propagation.

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Electric Circuits and Resistance

1. Ohm’s Law for Circuits:

   • The relationship between current, voltage, and resistance in a circuit: $V = IR$

2. Power in Electrical Circuits:

   • The rate at which electrical energy is used: $P = IV = I^2 R = \frac{V^2}{R}$

3. Resistors in Series and Parallel:

   • Series: $R_{\text{total}} = R_1 + R_2 + \dots + R_n$
   • Parallel: $\frac{1}{R_{\text{total}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$

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Numerical Examples

1. Example 1: A coil with a resistance of $10 \, \Omega$ carries a current of $2 \, A$. Find the power dissipated in the coil.

   • Formula: $P = I^2 R$
   • Substituting values: $P = 2^2 \cdot 10 = 40 \, W$

2. Example 2: A long straight wire carries a current of $5 \, A$. Find the magnetic field at a distance of $0.2 \, m$ from the wire.

   • Formula: $B = \frac{\mu_0 I}{2 \pi r}$
   • Substituting values: $B = \frac{4\pi \times 10^{-7} \cdot 5}{2 \pi \cdot 0.2} = 5 \times 10^{-6} \, T$

3. Example 3: A capacitor of capacitance $10 \, \mu F$ is charged to $12 \, V$. Calculate the energy stored in the capacitor.

   • Formula: $E = \frac{1}{2} C V^2$
   • Substituting values: $E = \frac{1}{2} \cdot 10 \times 10^{-6} \cdot 12^2 = 0.00072 \, J$

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Recap: Key Points to Remember

• Electricity involves the flow of electric charge, governed by Ohm’s law, Kirchhoff’s laws, and other principles.
• Magnetic fields are created by currents, and changes in these fields induce electric currents through electromagnetic induction.
• Power in electrical circuits is related to current, voltage, and resistance.
• Electromagnetic waves are a combination of electric and magnetic fields, traveling through space at the speed of light.