Surface Charge Density Formula

Surface Charge Density Formulas & Examples

Surface charge density is a measure of how much electric charge is accumulated over a surface. It is calculated as the charge per unit surface area.

If q is the charge and A is the area of the surface, then the surface charge density is given by; \[\sigma = \frac{q}{A}\],
The SI unit of surface charge density is Cm–2.

Example 1
1. A large plane sheet has an area 50 cm2 and has a charge of 3mC uniformly distributed over it. Find the surface charge density.
Solution:
q = 3 mC = 3 × 10–3 C, A = 50 cm2 = 5 ×10–3 m2, \[\sigma \] = ?
\[\sigma = \frac{q}{A} = \frac{{3 \times {{10}^{--3}}}}{{5 \times {{10}^{--3}}}} = 0.6\,C{m^{--2}}\]

2. A cuboidal box penetrates a huge plane sheet of charge with uniform surface charge density 2.5×10–2 Cm–2 such that its smallest surfaces are parallel to the sheet of charge. If the dimensions of the box are 10 cm × 5 cm × 3 cm, then find the charge enclosed by the box.
Solution:
Charge enclosed by the box = charge on the portion of the sheet enclosed by the box.
The area of the sheet enclosed; A = area of the smallest surface of the box
= 5 cm × 3 cm = 15 cm2 = 15 × 10–4 m2
Charge density; \[\sigma \]= 2.5 ×10–2 Cm–2
Charge enclosed; \[q = \sigma A = 2.5 \times {10^{--2}} \times 15 \times {10^{--4}} = 37.5 \times {10^{--6}}C = 37.5\,\mu C\]

Practice question:
The same charge is given to four thin plane laminas of different shapes; an equilateral triangle, a square, a regular hexagon and a circular one. All of these have the same perimeter. Then the lamina with the maximum surface charge density has the shape as that of :
(a) an equilateral triangle (b) a square (c) a regular hexagon (d) a circle
Ans (a)