Conductors & Electrostatic Shielding

Electrostatics of Conductors

A conductor contains free charges that can move easily inside it. In metals, these free charges are electrons.

When a conductor is placed in an electric field, or when excess charge is given to it, these free charges begin to move under the action of electric force.

This movement continues until a final state is reached in which:

• no further motion of free charges takes place,
• the net electric force on every free charge inside the conductor becomes zero.

This final state is called electrostatic equilibrium.

This leads to six fundamental properties that are essential for solving advanced electrostatics problems.

Six Fundametal Properties

1. Inside a conductor, the electrostatic field is zero

In a static situation, if there were an electric field inside, it would exert force on the free electrons, causing them to move. They would move until they redistribute themselves in such a way that their own field perfectly cancels the external field. Thus, the net field inside is always zero.

2. At the surface, the electric field must be normal to the surface

If the field had a tangential component, electrons on the surface would experience a force and start flowing (creating a surface current). Since we are in 'electrostatics' (charges at rest), the tangential component must be zero. Therefore, the field is always perpendicular to the surface.

3. The interior has no excess charge

By Gauss’s Law, since the electric field ($E$) is zero everywhere inside, the flux through any internal surface is zero. This means the net charge enclosed ($q_{in}$) must be zero. Any excess charge given to a conductor must reside entirely on its outer surface.

4. Electrostatic potential is constant throughout the volume

Since $E = 0$ inside, the potential gradient $dV/dr$ is also zero. This means the potential $V$ does not change from point to point.

• Key Fact: The potential at any point inside a conductor is exactly equal to the potential on its surface.

5. Electric field at the surface of a charged conductor

The magnitude of the electric field at the surface of a conductor with surface charge density $\sigma$ is: $$E = \frac{\sigma}{\epsilon_0}$$ In vector form: $\vec{E} = \frac{\sigma}{\epsilon_0} \hat{n}$, where $\hat{n}$ is the unit vector normal to the surface.

6. Electrostatic Shielding

This is a fascinating application of property #1. If a conductor has a cavity (a hole) inside it, the electric field inside that cavity is zero, even if the conductor is placed in a massive external field or is highly charged. This 'safe zone' is called Electrostatic Shielding.

🧠 Memory Capsule

• Internal Field: $E = 0$ (The most fundamental rule).
• Charge Location: Only on the outer surface. None inside.
• Surface Field: Always $90^{\circ}$ to the surface; Magnitude $= \sigma/\epsilon_0$.
• Potential: Constant everywhere inside ($V_{in} = V_{surface}$).
• Shielding: Cavities in conductors are protected from external electrical influences. Think of a car in a lightning storm!

Example 1: Potential at the Center

A hollow metallic sphere of radius 20 cm is given a charge of $+5 \mu C$. What is the electric potential at its center?

Solution:

1. Identify the Rule: For a conductor, potential is constant throughout. $V_{center} = V_{surface}$.
2. Calculate Surface Potential: $V = kQ/R = (9 \times 10^9 \times 5 \times 10^{-6}) / 0.2$.
3. Simplify: $V = 4.5 \times 10^4 / 0.2 = 2.25 \times 10^5$ V.
4. Result: The potential at the center is $2.25 \times 10^5$ V.

Example 2: Field inside a Cavity

A conductor has a cavity with a charge $+q$ suspended inside it (not touching). What is the charge on the inner and outer surfaces of the conductor?

Solution:

1. Apply Gauss Law: The field inside the metal must be zero. To cancel the $+q$ inside the cavity, a charge of $-q$ must be induced on the inner wall of the cavity.
2. Conservation of Charge: If the conductor was initially neutral, and $-q$ moved to the inner wall, a charge of $+q$ must appear on the outer surface.
3. Result: Inner surface: $-q$; Outer surface: $+q$.

Example 3: Work Done inside a Conductor

How much work is required to move a $2 \mu C$ charge from the surface of a charged conductor to a point 5 cm deep inside it?

Solution:

1. Analyze Potential: The potential is constant throughout the volume of the conductor.
2. Potential Difference: $\Delta V = V_{inside} - V_{surface} = 0$.
3. Calculate Work: $W = q \Delta V = 2 \mu C \times 0 = 0$.
4. Result: Work done is zero.

Example 4: Surface Field Magnitude

The surface charge density on a copper sphere is $8.85 \times 10^{-8} \text{ C/m}^2$. Calculate the electric field just outside the surface.

Solution:

1. Use Formula: $E = \sigma / \epsilon_0$.
2. Identify Constants: $\epsilon_0 = 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2$.
3. Calculate: $E = (8.85 \times 10^{-8}) / (8.85 \times 10^{-12})$.
4. Simplify: $E = 10^4$ N/C.
5. Result: $10,000$ N/C.

Example 5: Shielding Logic (Conceptual)

Why do sensitive electronic instruments often come in metallic boxes?

Solution:

1. Principle: Electrostatic Shielding.
2. Explanation: The metallic box acts as a conductor. Any external electric field causes charges on the box to redistribute, ensuring the field inside the box remains zero.
3. Benefit: This protects the delicate internal circuits from external electrical noise or static interference.

Example 6: Net Charge on a Shell

A spherical conducting shell A has charge $q$ and another shell B has charge $2q$. They are connected by a wire. What is the final field inside shell A?

Solution:

1. Redistribution: When connected, charges flow until potentials are equal. However, for any conductor (or connected system of conductors), all excess charge resides on the outermost surface.
2. Final State: All charges ($3q$) move to the outer surface of shell B.
3. Inside A: Since there is no charge inside or on shell A, the field $E$ inside A is zero.

Example 7: Induced Charge near a Plate

A point charge $+Q$ is placed near an uncharged large conducting plate. What is the net charge on the plate?

Solution:

1. Induction: The $+Q$ charge attracts electrons to the near side and repels positive charges to the far side.
2. Net Charge: Since no charge was added or removed from the plate, the net charge remains zero, although it is now 'polarized'.

Example 8: Earthing a Conductor

A positively charged conductor is connected to the Earth. Describe the flow of charge.

Solution:

1. Potential Difference: The conductor is at a high (positive) potential, while Earth is at zero potential.
2. Flow: Electrons (negative) will flow from the Earth to the conductor to neutralize the positive charge.
3. Result: The conductor becomes neutral and its potential becomes zero.

Example 9: Comparison of Potential

Two metallic spheres, one hollow and one solid, have the same radius. Both are given the same charge $Q$. Which one will be at a higher potential?

Solution:

1. Charge Distribution: In both cases (hollow or solid), the charge $Q$ resides entirely on the outer surface.
2. Potential Formula: $V = kQ/R$.
3. Result: Since $Q$ and $R$ are the same, both spheres will have the same potential.

Example 10: Potential inside a non-uniform conductor

A conductor of an irregular shape is given a charge. Is the potential same at a sharp tip and a flat surface?

Solution:

1. Rule: In electrostatic equilibrium, the entire conductor is an equipotential body.
2. Result: Yes, the potential is the same everywhere on and inside the conductor.
3. Note: While potential is the same, the charge density $\sigma$ and electric field $E$ are much higher at the sharp tips!