3. ELECTROSTATICS AND CURRENT ELECTRICITY

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3. ELECTROSTATICS AND CURRENT ELECTRICITY

This chapter deals with the fundamental principles of static electric charges, electric fields, and how electric current behaves in circuits. It explains how charges interact and how current flows in conductors.

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3.1 Coulomb's Law, Unit of Charge

Coulomb's Law: Coulomb’s law describes the force between two point charges. The force ($F$) between two charges $q_1$ and $q_2$ is directly proportional to the product of the charges and inversely proportional to the square of the distance ($r$) between them.

$$F = k_e \cdot \frac{q_1 q_2}{r^2}$$

Where:

• $F$ is the electrostatic force between the charges.
• $k_e$ is Coulomb’s constant, $k_e=9\times {10}^9{\text{N m}}^2\mathrm{/}{\text{C}}^{2.}$
• $q_1$ and $q_2$ are the magnitudes of the two point charges.
• $r$ is the distance between the charges.

Unit of Charge:

• The SI unit of charge is the Coulomb (C).
• 1 Coulomb is the amount of charge that passes through a conductor carrying a current of 1 ampere for 1 second.

Example:
If two charges $q_1 = 5 \, \mu C$ and $q_2 = 3 \, \mu C$ are 0.2 meters apart, the force between them can be calculated using Coulomb’s law.

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3.2 Electric Field, Electric Lines of Force and Their Properties

Electric Field (E):

• An electric field is a region around a charged particle where another charged particle experiences a force. It is defined as the force $F$ experienced by a small positive test charge $q_0$ placed in the field.

  $$E = \frac{F}{q_0}$$

• Unit: The unit of electric field is N/C (Newton per Coulomb).

• The direction of the electric field is the direction in which a positive test charge would move.

Electric Lines of Force:

• Electric lines of force are the paths along which a positive test charge would move under the influence of the electric field.
• They always start at positive charges and end at negative charges.

Properties of Electric Lines of Force:

1. They never intersect.
2. They are closer together where the electric field is stronger.
3. They begin at positive charges and end at negative charges.
4. They are directed away from positive charges and towards negative charges.
5. They are three-dimensional, not just confined to a plane.

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3.3 Electric Flux

• Electric Flux (Φ):
  Electric flux represents the quantity of electric field passing through a given area. It depends on the electric field and the area through which the field lines pass.

  $$\Phi_E = E \cdot A \cdot \cos(\theta)$$
  

  Where:

  • $\Phi_E$ is the electric flux.
  • $E$ is the electric field strength.
  • $A$ is the area through which the electric field lines pass.
  • $\theta$ is the angle between the electric field and the normal to the surface.

Unit: The unit of electric flux is Nm²/C.

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3.4 Electric Current and its Units

Electric Current (I):

• Electric current is the rate of flow of electric charge through a conductor.

  $$I = \frac{Q}{t}$$

  Where:

  • $I$ is the current (in amperes, A).
  • $Q$ is the total charge passing through the conductor (in coulombs).
  • $t$ is the time taken (in seconds).

Unit of Electric Current:

• The unit of electric current is the Ampere (A). 1 Ampere is defined as 1 coulomb of charge passing through a conductor per second.

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3.4.1 Direct and Alternating Current

• Direct Current (DC):
  In DC, the electric charge flows in a single direction. It is used in most electronic devices like batteries, flashlights, and mobile phones.

  Example: The current from a battery is DC, as the charge flows from the negative terminal to the positive terminal.

• Alternating Current (AC):
  In AC, the direction of the electric charge reverses periodically. AC is used in most household appliances, and the voltage alternates in direction, typically at a frequency of 50 Hz or 60 Hz depending on the country.

  Example: The electricity supplied to homes is AC.

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3.5 Kirchhoff’s Laws

Kirchhoff’s Current Law (KCL):

• KCL states that the total current entering a junction in an electric circuit is equal to the total current leaving the junction.

  $$\sum I_{\text{in}} = \sum I_{\text{out}}$$

Kirchhoff’s Voltage Law (KVL):

• KVL states that the sum of the potential differences (voltages) around any closed loop in a circuit is equal to zero.

  $$\sum V = 0$$
  

Example: In a simple circuit with three resistors, KCL helps determine how current divides at junctions, and KVL helps calculate the voltage drop across resistors.

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3.6 Wheatstone Bridge and Its Applications (Meter Bridge)

Wheatstone Bridge:

• The Wheatstone Bridge is a circuit used to measure an unknown resistance by balancing two legs of a bridge circuit. It consists of four resistors, a galvanometer, and a battery.

  Formula:
  The balance condition of a Wheatstone Bridge is given by:

  $$\frac{R_1}{R_2} = \frac{R_3}{R_4}$$

  Where $R_1$, $R_2$, $R_3$, and $R_4$ are the resistances in the bridge, and $R_1$ and $R_2$ are known, while $R_3$ is the unknown resistance.

Meter Bridge:

• A meter bridge is a practical implementation of the Wheatstone Bridge, typically used in school labs for measuring resistances.
• The bridge is a one-meter-long wire of uniform cross-section. The resistors and the galvanometer are connected in a way that allows resistance measurement by finding the point where the galvanometer shows zero current.

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3.7 Concept of Terminal Potential Difference and Electro Motive Force (EMF)

Electromotive Force (EMF):

• The EMF of a source (like a battery) is the potential difference between the two terminals when no current is flowing. It represents the energy provided per unit charge to move charges through the circuit.

  Unit: The unit of EMF is Volts (V).

Terminal Potential Difference:

• The terminal potential difference is the potential difference across the terminals of a battery when a current is flowing. It is less than the EMF due to internal resistance of the battery.

  $$V_{\text{terminal}} = \text{EMF} - I \cdot r_{\text{internal}}$$

  Where $r_{\text{internal}}$ is the internal resistance of the battery and $I$ is the current.

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Summary and Key Concepts:

• Coulomb’s Law: Describes the force between two charges.
• Electric Field: The region around a charge where other charges experience a force.
• Electric Flux: The measure of the electric field passing through a surface.
• Electric Current: The flow of charge, measured in amperes.
• Direct vs Alternating Current: DC is constant; AC alternates direction.
• Kirchhoff’s Laws: Used to analyze complex circuits (current and voltage laws).
• Wheatstone Bridge: Used to measure unknown resistance.
• EMF vs Terminal Potential Difference: EMF is the voltage when no current flows; terminal potential difference is the voltage when current flows.

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Numerical Example on Electric Current:

Problem: A current of 2 A flows through a conductor for 5 minutes. How much charge flows through the conductor?

Solution:
Using the formula $I = \frac{Q}{t}$, rearrange to find $Q$:

$$Q = I \cdot t = 2 \, \text{A} \cdot (5 \, \text{minutes} = 5 \times 60 = 300 \, \text{seconds})$$
$$Q = 2 \times 300 = 600 \, \text{Coulombs}$$

Answer: 600 Coulombs of charge flow through the conductor.

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