Electrostatics of Conductors: We know that conductors contain mobile charge carriers. In metal, the valence electrons act as the charge carrier. The valence electrons are very loosely bonded with their atoms, and they are to move within the metal but can’t leave the metal. Electrons drift in the presence of an external electric field against the direction of the field, but the positive ions remain fixed inside the nuclei.

The bounded electrons also remain held with the fixed nuclei. In the case of electrolytic conductors, positive and negative ions act as charge carriers. In this case, the movement of the charge carriers is affected by the external electric field and the chemical forces. In this article, we will learn more about the behaviour of conductors when placed in the external electric field.

Electric Field inside a Conductor

Consider a conductor, neutral or charged or kept in an external electrostatic field. In the static situation, the electric field is zero everywhere inside the conductor (no movement of charged particles). It is one of the defining properties of a conductor. A conductor has electrons. As long as the electric field is not zero, the charge carriers would experience force. In the static situation, the charges will be distributed so that the magnitude of the electric field is zero everywhere inside the conductor.

Learn about Conductors & Insulators of Electricity here

Direction of the Electric Field at every point on the Surface

Let’s assume that electric field \((E)\) lines are not normal to the surface and are inclined at some angle. If we resolve the electric field, then it will have some non-zero components along the surface. charges on the conductor’s surface would then experience force along the surface and move along the surface. But in the static situation, electrons are not moving in any certain direction, or we can say they are not experiencing any force. Therefore,\(E\) should not have any tangential component. Thus, we can say that the electric field at every point on the surface of a charged conductor must be normal to the surface.

Distribution of Charge in Conductors

Positive and negative charges will be equally distributed in every small volume or surface element in a neutral conductor. When the conductor is supplied extra charge, the excess charges reside only on the surface in the static situation. To understand this, let us take any arbitrary volume element \(V\) inside a conductor. On the closed surface \(S\) bounding the volume element \(V\) the electrostatic field will be zero. Thus, the total electric flux passing through \(S\) will be zero. From Gauss’s law,  we can say that the net charge enclosed by \(S\) is zero. It means there is no net charge at any point inside the conductor, and any excess charge must reside at the conductor’s surface.

Electrostatic Potential in Conductors

The electrostatic potential is constant everywhere inside the volume of the conductor and has the same value (as inside) on its surface. As we know that the value of electric field \(E\) inside the conductor is zero, the electrostatic potential difference \(( – E.dr)\) will be zero. So, the electrostatic potential remains constant throughout the volume of the conductor.  Also, electric fields have no tangential component on the surface, so no work is required in moving a small test charge within the conductor and on its surface. That means the potential difference between any two points inside or on the conductor’s surface is zero. If the conductor is charged, the electric field will be normal to the surface; this means there will be a potential difference between the surface and a point just outside the surface.

Electric Field at the Surface of a Charged Conductor

Fig: The Gaussian surface

According to the gauss law,
\(E = \frac{\sigma }{\varepsilon }\hat n\quad  \ldots {\rm{ (1)}}\)
where \(\sigma \) is the surface charge density and \(\hat n\) is a unit vector normal to the surface in the outward direction.
To derive the result, choose a short cylindrical volume and take the Gaussian surface about any point \(P\) on the surface. As shown in the above figure, the cylinder should be chosen such that it is partly inside and partly outside the surface of the conductor. It has a small area of cross-section \(dS\) and negligible height. Just inside the surface, the electrostatic field is zero, and outside, the field is normal to the surface with magnitude \(E\) Thus, the contribution to the total flux through the short cylinder comes only from the outside (circular) cross-section of the short cylinder. This equals \( \pm EdS\) (positive for \(\sigma \,\, > \,\,0\), negative for \(\sigma \,\, < \,\,0\) Over the small area \(dS,\)\(E\) may be considered constant; also, \(E\) and \(dS\) are parallel or antiparallel. The charge enclosed by the short cylinder is \(\sigma dS\).
By Gauss’s law:
\(EdS = \frac{{|\sigma |dS}}{\varepsilon }\)
\(E = \frac{\sigma }{\varepsilon }\quad  \ldots (2)\)
Including the fact that the electric field is normal to the surface, we get the vector relation, Eq.\((1)\) which is true for both signs of \(\sigma \) For \(\sigma  > 0\) the electric field is normal to the surface in the outward direction; for \(\sigma  < 0\) he electric field is normal to the surface in the inward direction.

Electrostatic Shielding

The electric field inside a conductor cavity at any point is zero(if no charge is placed in the cavity). To understand this, let us take a conductor with a cavity. From Gauss law, the charge enclosed by the cavity is zero, then the electric field inside the cavity will be zero. We know that if the conductor is charged or charges are induced on a neutral conductor by an external field, all charges reside only on the outer surface with the cavity. In the presence of an external electric field, the electric field inside the cavity will always be zero. It does not depend on the size and shape of the cavity and the charge on the conductor. Any cavity in a conductor remains shielded from outside electric influence; the electric field inside the cavity will always be zero. This phenomenon is known as electrostatic shielding. This phenomenon is used for protecting sensitive instruments from outside electrical influence.

Solved Example on Electrostatics of Conductors

Q.1. A point charge \((q)\) is placed inside a cavity of the conductor, as shown. Another point charge \(Q\) is placed outside the conductor, as shown. Now, as the point charge \(Q\) is pushed away from the conductor. What will be the effect on the potential difference \(({V_A} – {V_B})\) between two points \(A\) and \(B\) within the cavity of the sphere?

Solution:- The electric field inside the cavity due to any external charge will always be zero. Any cavity in a conductor remains shielded from outside electric influence. So, inside the cavity, the electric field is only due to internal charge, and there will not be any effect of movement of outside charge. So, the potential difference \(\left( {{{{V}}_{{A}}} – {{{V}}_{\rm{B}}}} \right)\) will remain constant.

Summary

In the static situation, the net electric field is zero everywhere inside a conductor. The electrostatic field at the surface of a charged conductor must be normal to the surface at every point. There is no net charge at any point inside the conductor, and any excess charge will always reside on the surface. The potential difference between any two points inside or on the surface of the conductor is zero. The electric field at the surface of a charged conductor is given by,
\(E = \frac{\sigma }{n}\hat n\) where \(\sigma \) is the surface charge density and n is a unit vector normal to the surface in the outward direction. 
The electric field inside the cavity of any conductor will always be zero. This phenomenon is known as electrostatic shielding. In electrostatic shielding, any cavity in a conductor remains shielded from outside electric influence: This phenomenon is used for protecting sensitive instruments from outside electrical influence.

FAQs on Electrostatic of Conductors

Q.1. What will be the value of the electric field inside a solid iron sphere having a charge of \(4\,{\rm{C}}?\)
Ans: Iron is an electrical conductor. We know that the magnitude of the electric field inside any conductor will always be zero, so the electric field inside the ball will be zero.

Q.2. What is an equipotential surface?
Ans: The surface formed by the locus of all points having the same potential is known as the equipotential surface.

Q.3. What is electrostatic shielding?
Ans: Any cavity in a conductor remains shielded(unaffected) from outside electric influence; this is known as electrostatic shielding.

Q.4. Write down the few applications of electrostatic shielding?
Ans: Applications of electrostatic shielding are,
(a) Elevators act as an unintended Faraday cage as the signals from the phones and radios get shielded.
(b) To avoid electrocution, electrical linemen wear suits that are made of Faraday cages.
(c) Used for protecting sensitive instruments from outside electrical influence.

Q.5. How much work is required to move a point charge from centre to surface of the charged hollow sphere of copper?
Ans: As we know, the electric field inside the hollow sphere(conductor) will be zero. So the potential difference between the centre and any point at the surface will be zero. So, the work done will be zero.

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