Capacitance of Parallel Plate Capacitor

The diversity in the variety of capacitors is astonishing.

Starting from the cylindrical shaped capacitors.

Through flat bead like looking capacitors.

To parallel plate capacitors.

The parallel plate capacitors are the most widely used ones.

Let’s look at the structure of the parallel plate capacitor.

The parallel plate capacitor is made of two plates separated by a distance and parallel to each other.

Here one plate will be positively charged and the other one will be negatively charged.

Let’s put some charge on the plates and see the charge distribution.

We are giving +q charge to plate ‘A’ and -q charge to plate ‘B’

Now to see the charge distribution we are dividing the capacitor in four regions numbered 1,2,3,4

Now the charge in region 1 would be zero.

Now the charge in region 4 would be zero.

Now the charge in region 2 would be +q

Now the charge in region 3 would be -q.

Let’s look at the electric field in those four regions.

One thing to remember while calculating the electric field is that the electric field always points from positive to negative charges.

The electric fields in region 1 due to +ve and _ve charge are E2 and E3. The magnitudes of the E2 and E3 are same but the directions are opposite.

The total electric field in region 1 is 0 as E2 and E3 with equal magnitudes cancels each other.

The electric fields in region 4 due to +ve and _ve charge are E2 and E3.The magnitudes of the E2 and E3 are same but the directions are opposite.

The total electric field in region 4 is 0 as E2 and E3 with equal magnitudes cancels each other

The electric fields in region 2 and 3 due to +ve and _ve charge are E2 and E3 again.The magnitudes the directions of the E2 and E3 are same.

The total electric field in region 2,3 will be sum of E2 and E3 as they are in the same direction.

Let’s calculate the magnitude of the electric in between the plates.

We consider surface charge density to calculate the electric field using gauss's law.

The magnitude of E2 is

The magnitude of E3 is

Total electric field in between the plates(region 2 and 3) E is

Now let’s get the potential difference between the plates from the electric field

Relation between the electric field and potential difference is

The potential difference between the plates ,V is

Finally let’s calculate the capacitance of the parallel plate capacitor.

We know capacitance is accumulated charge per unit potential difference.

So in this case capacitance,C is

The potential of two conductor system is less than one conductor

The capacitance depends on the area of the plates and the separation between them.

If the cross sectional area of the plates increases the capacitance will increase

But if the separation between the plates increases the capacitance will decrease.

Revision

The parallel plate capacitor is made of two parallel plates with opposite charges.We have +q and -q charges only on the inner side of the plates.

There is electric field only between the plates.From the electric field we get the potential difference between the plates.

Form the ratio of charge and potential difference we get the the capacitance

The capacitance of the parallel plate capacitors is proportional to the cross sectional area of the plates and inversely proportional to the separation of the plates.

The End