Electrostatics of Conductors

 Electrostatics of Conductors

    Conductors have mobile charge carriers in metals, these are electrons. Valence electrons in metals detach from atoms and move freely, creating a "gas" of electrons within the metal that can't escape. They randomly collide but drift against an external electric field. In electrolytic conductors, both positive and negative ions carry charge, influenced by electric fields and chemical forces. This summary focuses on metallic conductors and their electrostatic behavior.

Inside a conductor, the electrostatic field is zero

Inside a conductor, the electrostatic field is zero, whether it’s neutral or charged, even with an external electrostatic field. In a static condition with no current flow, there is no electric field inside the conductor. This defines a conductor. A conductor has free electrons. If an electric field existed, free electrons would experience force and move. In a static condition, the free charges redistribute to ensure there’s no electric field every inside the conductor.

At the surface of a charged conductor, the electrostatic field must be normal to the surface at every point

    The electrostatic field at the surface of a charged conductor must be perpendicular to the surface. If it weren’t, there would be a tangential component causing free charges on the surface to move, disrupting the static condition. Therefore, the electric field at the surface must be normal to avoid charge movement along the surface.

The interior of a conductor can have no excess charge in the static situation

In a static situation, the interior of a conductor contains no excess charge. A neutral conductor has equal amounts of positive and negative charges throughout. When charged, any excess charge resides on the surface, not within the conductor. Gauss’s law explains that for any volume inside a conductor, the electric field on the surrounding surface is zero, leading to zero electric flux and thus no enclosed charge. Since this applies even to arbitrarily small volumes, excess charge must be on the surface, not inside.

The electrostatic potential is constant throughout the volume of the conductor and has the same value (as inside) on it's surface

Since the electric field inside a conductor is zero and has no tangential component on its surface, moving a test charge within or across the conductor’s surface requires no work (W = 0). Consequently, there’s no potential difference between any points within or on the surface. However, a charged conductor will have a normal electric field, indicating a potential difference between the surface and just outside it. In a system of conductors with varying size, shape, and charge, each conductor has a constant potential, but these values may vary among conductors.

The electric field at the surface of a charged conductor `\vecE = \frac{\sigma}{\epsilon_0}\hat n`

Consider a Pillbox at any point on the conductor’s surface. It has a negligible height small cross-sectional area dA.

The electric field is zero inside the conductor, but just outside, it is normal to the surface with a magnitude E. Thus the contribution to the total flux through the pill box comes only from the outside cross-section of the pill box.

According to Gauss’s law

`E\ dA\ =\ \frac{q}{\varepsilon_0}`

`E\ dA\ =\ \frac{\sigma\ dA}{\varepsilon_0}`

`E\ =\ \frac{q}{\varepsilon_0}`

Electric field is normal to the surface.

If `\sigma\ >\ 0` then the electric field is normal to the surface outward to the surface.

If `\sigma\ <\ 0` then the electric field is normal to the surface inward to the surface.

The electric field direction is normal to the surface – outward for positive charge and inward for negative charge.

Electrostatic shielding

A conductor has no electric field inside it, even in an external electric field. This happens because the free electrons in the conductor rearrange themselves to cancel out the external field, creating an equilibrium state. The electron movement creates an internal charge distribution that opposes the external field.

Electrostatic Shielding

It is clear that the cavity within a conductor remains unaffected by external electric fields. Even if the conductor has a charge or is influenced by external fields, all charges reside on the outer surface, protecting any equipment or sensitive instruments within the cavity.

Applications and Uses of Electrostatic shielding

Electrostatic shielding is commonly used in various applications to protect sensitive electronic components, such as circuits and communication devices.

Example –

Why a Comb Attracts Paper After Running Through Dry Hair

Running a comb through dry hair creates static electricity from friction. This transfers electrons to the comb, giving it a negative charge. The charged comb then attracts neutrally charged paper bits due to induced positive charges. 

Wet hair or rainy weather reduces static. Moisture conducts electricity, letting charges dissipate, so combs can’t attract paper bits.

Example –

Why are aircraft tires slightly conductive when ordinary rubber is typically an insulator?

Aircraft tires are made slightly conducting to prevent static electricity build-up. When aircraft land or take off, they generate significant friction with the runway, causing static charges to accumulate on the aircraft and its tires If the tires were fully insulating, this static electricity could build to dangerous levels, potentially leading to sparks of electrical discharges.

By making the tires slightly conductive, the static charges can safely dissipate into the ground through the tires, reducing the risk of static discharge-related hazards. This design is crucial for ensuring safe ground operations and preventing possible ignition sources around aircraft fuel and other sensitive components.

Vehicles carrying inflammable materials usually have metallic ropes touching the ground during motion. Why?

Vehicles transporting flammable materials often have metallic ropes touching the ground to prevent static electricity build-up. As the vehicle moves, friction can generate static charges, which pose a risk of ignition. The metallic ropes conduct static charges to the ground, dissipating them safely, thereby reducing the risk of sparks or fire.

A bird perches on a bare high power line, and nothing happens to the bird. A man standing on the ground touches the same line and gets a fatal shock. Why?

A bird on a power line is safe because it doesn’t create a path for electricity to the ground. A person touching the line while standing on the ground creates a complete circuit, allowing electricity to flow through them to the ground, leading to a fatal shock.

Why are aircraft tires slightly conductive when ordinary rubber is typically an insulator?

Aircraft tires are slightly conductive to avoid static build-up from friction during takeoff and landing, which could cause dangerous sparks or electric discharges.

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