A cylindrical capacitor is made of two thin-walled concentric cylinders. The inner cylinder has radius 4mm , and the outer one a radius 13mm . The common length of the cylinders is 153m . What is the potential energy stored in this capacitor when a potential difference 7V is applied between the inner and outer cylinder? (k =1/4∗π∗ϵ0=8.99×109N⋅m2/C2)
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capcyl.html
determine the capacitance from that.
Then, energy=1/2 C V^2
To calculate the potential energy stored in the capacitor, we can use the formula:
Potential energy (U) = 1/2 * (C * V^2)
Where:
- C is the capacitance of the capacitor
- V is the potential difference applied
To find the capacitance, we can use the formula:
C = (2πϵ0 * L) / ln(b/a)
Where:
- ϵ0 is the permittivity of free space (given as k * 1/4πϵ0)
- L is the length of the cylinders
- a is the radius of the inner cylinder
- b is the radius of the outer cylinder
Let's plug in the given values and solve step by step:
ϵ0 = 8.99 × 10^9 N⋅m^2/C^2 (given)
k = 1 / (4πϵ0) = 1 / (4 * π * 8.99 × 10^9)
L = 153 m (given)
a = 4 mm = 0.004 m (given)
b = 13 mm = 0.013 m (given)
V = 7 V (given)
1. Calculate k:
k = 1 / (4 * π * 8.99 × 10^9)
2. Calculate the capacitance (C):
C = (2πϵ0 * L) / ln(b/a)
3. Substitute the given values into the equation to find C.
4. Calculate the potential energy (U):
U = 1/2 * (C * V^2)
Now, let's calculate each step step-by-step.
To find the potential energy stored in a cylindrical capacitor, you'll need to use the formula:
U = (1/2) * C * V^2
Where U is the potential energy, C is the capacitance, and V is the potential difference.
The capacitance of a cylindrical capacitor can be calculated using the formula:
C = (2 * π * ϵ0 * L) / ln(b/a)
Where C is the capacitance, L is the length of the cylinders, a is the radius of the inner cylinder, b is the radius of the outer cylinder, and ϵ0 is the permittivity of free space.
Given values:
a = 4 mm = 4 x 10^-3 m
b = 13 mm = 13 x 10^-3 m
L = 153 m
V = 7 V
ϵ0 = 8.99 x 10^9 N⋅m^2/C^2
First, convert the radii of the cylinders to meters:
a = 4 x 10^-3 m
b = 13 x 10^-3 m
Next, calculate the capacitance using the given formula:
C = (2 * π * ϵ0 * L) / ln(b/a)
C = (2 * π * 8.99 x 10^9 N⋅m^2/C^2 * 153 m) / ln(13 x 10^-3 m / 4 x 10^-3 m)
Simplify the expression:
C = (2 * π * 8.99 x 10^9 N⋅m^2/C^2 * 153 m) / ln(3.25)
Use a calculator to find the natural logarithm of 3.25:
ln(3.25) ≈ 1.178
Substitute the values into the formula:
C = (2 * π * 8.99 x 10^9 N⋅m^2/C^2 * 153 m) / 1.178
Calculate the capacitance:
C ≈ 6.22 x 10^-9 F
Now, plug the values of C and V into the potential energy formula:
U = (1/2) * C * V^2
U = (1/2) * 6.22 x 10^-9 F * (7 V)^2
Simplify and calculate the expression:
U = (1/2) * 6.22 x 10^-9 F * 49 V^2
U ≈ 1.52 x 10^-8 J
Therefore, the potential energy stored in this cylindrical capacitor when a potential difference of 7V is applied is approximately 1.52 x 10^-8 J.