Introduction to Electrostatics

                                                 ELECTROSTATICS 

🎓 ELECTROSTATICS INTRODUCTION IN SHORT

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🔌 1. Sources and Effects of Electromagnetic Fields

Electric and magnetic fields are produced by electric charges. A stationary charge produces an electric field, while a moving charge (like in a wire) produces both electric and magnetic fields. These fields influence other charges nearby.

Example: When you rub a balloon on your hair, electrons transfer, and the balloon sticks to a wall. The balloon creates an electric field that pulls on wall charges.

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📍 2. Coordinate Systems

To describe fields and charges in space, we use coordinate systems:

• Cartesian (x, y, z): For rectangular objects.

• Cylindrical (r, ϕ, z): For wires, cylinders.

• Spherical (r, θ, ϕ): For spheres, point charges.

Example: For a long, thin wire with uniform charge, cylindrical coordinates make it easier to calculate electric fields.

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🧭 3. Vector Fields – Gradient, Divergence, Curl

Electric fields are vector fields — they have both magnitude and direction.

• Gradient (∇V): Shows how fast electric potential changes with distance.

  $$\vec{E} = -\nabla V$$

• Divergence (∇·E): Measures how much a field spreads out. If divergence is positive, charges are present.

• Curl (∇×E): Measures rotation in the field.
  In electrostatics:

  $$\vec{\nabla} \times \vec{E} = 0$$

  (electric fields don’t swirl or rotate)

Example: If a metal ball is charged, the electric field radiates outward — this is divergence in action.

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⚖️ 4. Coulomb’s Law

Coulomb’s law gives the force between two point charges:

$$F = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q_1 q_2}{r^2}$$

Where:

• $q_1, q_2$ are the charges

• $r$ is the distance between them

• $\varepsilon_0$ is permittivity of free space

Example: Two charges of +2 µC placed 1 m apart will repel each other with a force of:

$$F = \frac{9 \times 10^9 \cdot (2 \times 10^{-6})^2}{1^2} = 36 \, \text{N}$$

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⚡ 5. Electric Field Intensity (EFI)

EFI is the force per unit charge at a point in space:

$$\vec{E} = \frac{\vec{F}}{q}$$

It shows the direction and strength of electric influence on other charges.

EFI for Different Charge Distributions:

a) Line Charge ($\lambda$):

$$\vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{\lambda \, dl}{R^2} \hat{R}$$

b) Surface Charge ($\sigma$):

$$\vec{E} = \frac{1}{4\pi\varepsilon_0} \iint \frac{\sigma \, dA}{R^2} \hat{R}$$

c) Volume Charge ($\rho$):

$$\vec{E} = \frac{1}{4\pi\varepsilon_0} \iiint \frac{\rho \, dV}{R^2} \hat{R}$$

Example: For an infinite line charge with $\lambda = 5 \, \mu C/m$, the electric field at distance $r$ is:

$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$

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🔋 6. Work Done in Moving a Charge

When a charge moves in an electric field, work is done:

$$W = q (V_A - V_B)$$

This work is stored as electric potential energy.

Example: If a 2 µC charge moves from 12 V to 2 V,

$$W = 2 \times 10^{-6} \cdot (12 - 2) = 20 \, \mu J$$

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🌐 7. Electric Potential (V)

Potential is the electric energy per unit charge:

$$V = \frac{1}{4\pi\varepsilon_0} \cdot \frac{q}{r}$$

For distributed charges, we integrate:

• Line: $V = \frac{1}{4\pi\varepsilon_0} \int \frac{\lambda \, dl}{R}$

• Surface: $V = \frac{1}{4\pi\varepsilon_0} \iint \frac{\sigma \, dA}{R}$

• Volume: $V = \frac{1}{4\pi\varepsilon_0} \iiint \frac{\rho \, dV}{R}$

Example: Potential 2 meters from a 3 µC point charge:

$$V = \frac{9 \times 10^9 \cdot 3 \times 10^{-6}}{2} = 13.5 \, \text{kV}$$

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📉 8. Potential Gradient

Electric field is related to the potential by:

$$\vec{E} = -\nabla V$$

It points from high to low potential.

Example: In a wire, voltage drops from 10 V to 0 V — the electric field drives current along this drop.

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🌍 9. Gauss’s Law

Gauss's law relates the total electric flux from a closed surface to the charge inside:

$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{inside}}}{\varepsilon_0}$$

This simplifies field calculation when there’s symmetry (sphere, cylinder, plane).

Applications:

a) Infinite Line Charge:

$$E = \frac{\lambda}{2\pi\varepsilon_0 r}$$

b) Infinite Sheet of Charge:

$$E = \frac{\sigma}{2\varepsilon_0}$$

c) Spherical Shell (outside):

$$E = \frac{Q}{4\pi\varepsilon_0 r^2}$$

Example: Field outside a charged sphere (like a metal ball) is just like from a point charge at its center.

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📘 10. Maxwell’s First Law (Electrostatics)

This is the differential form of Gauss’s Law:

$$\nabla \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$$

It tells us that the electric field comes from charge. Where there is more charge, the field diverges more.

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🔢 Numerical Example Summary

Problem Type	Formula Used	Example
Force between charges	Coulomb’s Law	$F = \frac{q_1 q_2}{4\pi\varepsilon_0 r^2}$
Electric field due to line	$E = \frac{\lambda}{2\pi\varepsilon_0 r}$	Field from a wire
Electric potential	$V = \frac{q}{4\pi\varepsilon_0 r}$	Voltage near a charge
Work done	$W = q (V_A - V_B)$	Energy to move charge

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✅ SUMMARY

Electrostatics is the foundation of electric field theory. It helps us understand how charges interact, how fields are formed, and how energy is stored or transferred in electric systems. Whether it's a balloon sticking to a wall, a capacitor in a circuit, or the design of a satellite, electrostatics is at work everywhere.

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📌 Related Topics

• Class 12 Physics: Electric Charge & Energy
• Electrostatics – Introduction
• P-N Junction Diode
• PCM in Digital Communication