A balloon of radius 30.0 cm is sprayed with a metallic coating so that the surface is conducting. A charge of 8.85×10−8 C is placed on the surface. What is the potential on the balloon's surface?
2.65×10^3 V
Suppose that the some air is let out of the balloon, so that the radius shrinks to 13.5 cm, what is the new potential on the balloon's surface?
5.89×10^3 V
What is the change in electrical potential energy during the shrinking process? For your own interest, what does the work needed to change the potential energy?
i got the first two, need help on the third!
To find the change in electrical potential energy during the shrinking process, we need to first understand that the electrical potential energy is given by the equation:
U = (1/2) Q^2 / C
where U is the electrical potential energy, Q is the charge, and C is the capacitance.
We can use the formula for the capacitance of a conducting sphere to calculate the value of C. The formula is:
C = 4πε₀r
where ε₀ is the permittivity of free space (ε₀ = 8.85 × 10^(-12) C²/(N·m²)), and r is the radius of the sphere.
For the initial situation, the radius of the balloon is 30.0 cm, so r = 30.0 cm = 0.3 m. The charge is 8.85×10^(-8) C.
Plugging these values into the equation for capacitance:
C = 4πε₀r = 4π(8.85 × 10^(-12) C²/(N·m²))(0.3 m)
Calculating this value, we find that C ≈ 3.96×10^(-11) F.
Now we can calculate the initial electrical potential energy using the formula U = (1/2) Q² / C:
U_initial = (1/2)(8.85 × 10^(-8) C)² / (3.96 × 10^(-11) F)
Calculating this value, we find that U_initial ≈ 998.75 J.
Next, we need to find the new capacitance and the new electrical potential energy after the balloon shrinks. Since the charge remains constant, the change in electrical potential energy is given by the equation:
ΔU = U_final - U_initial
We can find U_final using the new radius of 13.5 cm (r = 0.135 m):
C_final = 4πε₀r = 4π(8.85 × 10^(-12) C²/(N·m²))(0.135 m)
Calculating this value, we find that C_final ≈ 7.19×10^(-12) F.
Using the formula U = (1/2) Q² / C, we can find the new electrical potential energy:
U_final = (1/2)(8.85 × 10^(-8) C)² / (7.19 × 10^(-12) F)
Calculating this value, we find that U_final ≈ 1.62×10^3 J.
Finally, we can calculate the change in electrical potential energy:
ΔU = U_final - U_initial = (1.62×10^3 J) - (998.75 J)
Calculating this value, we find that ΔU ≈ 621.25 J.
So, the change in electrical potential energy during the shrinking process is approximately 621.25 J.
Regarding the work needed to change the potential energy, the work done is equal to the change in electrical potential energy. In this case, the work done is approximately 621.25 J.