Formula for Surface Charge density of a conductor

What is charge density? Electric charge density is the electric charge per unit dimension. Electric charges can be distributed along the length, over the surface and in the volume of a conductor. In linear distribution, the charge distribution takes place along the length of the conductor. Similarly, the term Volume charge distribution implies the distribution of the charge in the volume of a conductor. In this article, we are going to discuss the definition and formula for Surface charge density of a conductor in the context of charge distribution on a conductor.

Contents in this article:

  1. Definition of Surface charge density
  2. Formula for surface charge density
  3. Unit of surface charge density
  4. Dimension of surface charge density
  5. Surface charge density of different shaped conductors
  6. Surface charge density of irregular shaped conductor
  7. How the surface charge density depends on the Curvature of the surface of the conductor?

Explanation of this article with a video

Video on Surface charge density

Definition of Surface charge density

Surface charge density of a conductor is defined as the amount of charge distributed per unit surface area of the conductor. It is denoted by the Greek letter sigma ({\color{Blue} \sigma }). One cannot obtain surface charge density in a very thin linear conductor. Apart from that, surface charge distribution exists on every charged conductor.

Formula for Surface charge density

If Q amount charge is distributed over the surface of a conductor of total surface area A, then the formula for the surface charge density of the conductor is \small {\color{Blue} \sigma =\frac{Q}{A}}

Different conductors of same charge can have different values of surface charge density if their surface area is different. We have discussed this below in surface charge density formula section.

Unit of Surface Charge density

Unit of Surface charge density = Unit of charge/unit of area

Now, the SI unit of charge is Coulomb (C) and the SI unit of surface area is m2 .

So, the SI unit of surface charge density is C/m2 .

Similarly, one can find that the CGS unit of surface charge density is esu/cm2 .

Dimension of surface charge density

The dimension of electric charge is [TI] and the dimension of surface area is [L2].

So, the dimensional formula of Surface charge density of a conductor is [ L-2 TI ].

Surface charge density formula of conductors of different shapes

The conductors of different shape have different surface area. So, they will have different values of surface charge density. Sphere or spherical shell, Cylinder, Capacitor all have different charge densities on their surface.

Formula for Surface Charge density of a conducting sphere

Let a conducting sphere of radius r have total charge Q on its surface. Now, the surface area of the sphere is A=4πr2.

So, the surface charge density is \small {\color{Blue} \sigma =\frac{Q}{4\pi r^{2}}}. This is the formula for surface charge density of a sphere.

Surface charge density of a conducting spherical shell is also same as that of a conducting sphere of same radius and same charge. Since, the curvature of the surface of a sphere is same at every point on its surface, the surface charge density is constant at everywhere on the surface of sphere.

Surface charge density of a charged Cylinder

We consider a cylindrical conductor of length L, radius r. Now, a cylinder has three surfaces – one curved surface and two flat surfaces. Surface charge density can be different for these two type of surfaces. Area of the curved surface is 2πrL and the surface area of each of the flat surfaces is πr2.

So, the surface charge density on the curved surface of the cylinder is \small {\color{Blue} \sigma = \frac{Q}{2\pi rL}} and the surface charge density on each flat surface of the cylinder is \small {\color{Blue} \sigma = \frac{Q}{\pi r^2}}.

Where, Q is the charge on the surface. The above two equations give the formula for surface charge density of a cylinder.

Surface Charge density of a Capacitor

Formula for surface charge density of a capacitor depends on the shape or area of the plates of the capacitor. If the capacitor consists of rectangular plate of length L and breadth b, then surface area is A=Lb. Then the surface charge density will be \small {\color{Blue} \sigma = \frac{Q}{Lb}}

If the plates of the capacitor have circular shape of radius r, then the surface charge density of the capacitor will be as \small {\color{Blue} \sigma =\frac{Q}{\pi r^{2}}}.

Surface charge density of a conductor of irregular shape

For a conductor of irregular surface, the surface area will be different at different points on its surface. So, surface charge density will vary point to point on its surface. The value of surface charge density will be greater at that region where the curvature is greater. That means it will have greater surface charge density at its edges. So, there is no permanent formula for surface charge density of an irregular shaped conductor.

How does the surface charge density vary with the curvature on the surface of the conductor?

Surface charge density depends on the curvature of the conductor. Surface charge density is greater if their is greater curvature. That means the edges of the surface will have greater charge density. This can be understand by using an example.

For a spherical conductor, the expression for surface charge density is \small {\color{Blue} \sigma =\frac{Q}{4\pi r^{2}}}. Now, greater curvature means lower radius. Then in this equation denominator becomes smaller for for greater curvature (smaller radius). So, the ratio i.e. the surface charge density increases with increase in curvature of the surface of the conductor.

Numerical problem:

Surface charge density of a spherical conductor of radius 10 cm is 0.7 C/m2. Find the total charge on its surface.

Surface charge density, { \sigma } = 0.7 C/m2

Radius of the sphere, r =10 cm = 0.1 m

So, Surface area of the sphere is A = 4πr2 = {4×3.14×(0.1)2}

or, A = 0.1256 m2

Thus, the total charge on the surface of the sphere is, Q = { \sigma }A

or, Q = (0.7×0.1256) = 0.8792

or, Q = 0.88 Coulomb (Up to two significant figures)

This is all from this article on surface density of charge. If you have any doubt to understand this topic, you can ask me in the comment section. Also, you can get physics and electronics related blog posts in this website. One can visit our website and access those blog articles for absolutely free.

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