What is the charge density? The electric charge density refers to the amount of 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 distribution of charge takes place along the length of the conductor. Similarly, the term Volume charge distribution implies the distribution of charges 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 over the surface of a conductor.

**Contents in this article:**

*Definition of Surface charge density**Formula for surface charge density**Unit of surface charge density**Dimension of surface charge density**Surface charge density of different shaped conductors**Surface charge density of irregular shaped conductor**How the surface charge density depends on the Curvature of the surface of the conductor?*

**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.

**Units 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 m^{2 }.

So, the SI unit of surface charge density is **C/m ^{2}** .

Similarly, one can find that the CGS unit of surface charge density is **esu/cm ^{2}**

^{ }.

**Dimension of surface charge density**

The dimension of electric charge is [**TI**] and the dimension of surface area is [**L ^{2}**].

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 shapes have different surface area. So, they will have different values of surface charge density. The forms of surface charge density equations are also different. 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=4πr ^{2}.**

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 equation 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 **πr**^{2}.

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/m**^{2}. Find the total charge on its surface.

^{2}. Find the total charge on its surface.

Surface charge density, { \sigma } = 0.7 C/m^{2}

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

So, the surface area of the sphere is A = 4πr^{2} = {4×3.14×(0.1)^{2}}

or, A = 0.1256 m^{2}

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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