A parallel plate capacitor with circular plates and a capacitance C has the radius of each plate doubled, and the distance between the plates doubled. What is the new capacitance?
A. C/2
B. C
C. 2C
D. 4C

To solve this problem, we need to understand the formula for the capacitance of a parallel plate capacitor.
Capacitance of a Parallel Plate Capacitor:
The capacitance CC of a parallel plate capacitor is given by the formula:
C=ε0AdC = \frac{\varepsilon_0 A}{d}
Where:
• ε0\varepsilon_0 is the permittivity of free space,
• AA is the area of each plate,
• dd is the distance between the plates.
For a circular plate, the area AA is given by:
A=πr2A = \pi r^2
Where rr is the radius of the plate.
Step 1: Change in the Radius of the Plate
The radius of the plate is doubled. So, the new radius is 2r2r.
The new area AnewA_{\text{new}} is:
Anew=π(2r)2=4πr2A_{\text{new}} = \pi (2r)^2 = 4\pi r^2
So, the area of the plate increases by a factor of 4.
Step 2: Change in the Distance Between the Plates
The distance between the plates is also doubled, so the new distance is 2d2d.
Step 3: New Capacitance
Now, let's calculate the new capacitance CnewC_{\text{new}} using the formula:
Cnew=ε0AnewdnewC_{\text{new}} = \frac{\varepsilon_0 A_{\text{new}}}{d_{\text{new}}}
Substitute the changes in area and distance:
Cnew=ε0⋅4A2d=2×ε0AdC_{\text{new}} = \frac{\varepsilon_0 \cdot 4A}{2d} = 2 \times \frac{\varepsilon_0 A}{d}
Since C=ε0AdC = \frac{\varepsilon_0 A}{d}, we get:
Cnew=2CC_{\text{new}} = 2C
Final Answer:
The new capacitance is 2C2C.
Answer: C. 2C