Use Gauss’ Law to write an equation for the electric field at a distance R1 < r < R2 from the center of the cylinders. Write your answer in terms of λ, r, and e0.
Problem 18: A cylindrical capacitor is made of two concentric
Part A: Use Gauss’ Law to write an equation for the electric field at a distance R1 < r < R2 from the center of the cylinders. Write your answer in terms of λ, r, and e0.
We want to start with the equation
For dA we can substitute since we need to use the lateral surface area of the cylinder and for q we can substitute since it would be the charge per unit length across the whole length of the cylinder leaving us the equation .
Part B: Calculate the electric potential difference between the outside and the inside cylinders in V.
For this we would want to use the equation we found in part a and integrate it with the limits R1 to R2 to get the difference between just the outside and inside cylinders, . After integrating we get E. Then plugging in the values, we get = 4.652E5 V
Part C: Calculate the capacitance per unit length of these concentric cylinders in F/m.
To find the capacitance per unit length we would use the equation with using the voltage we found in part B and as the charge (the l cancels out since it’s being divided by the length to find the capacitance per unit length). F/m
Part D: Calculate the energy stored in the capacitor per unit length, in units of J/m.
For this we would use the equation and plug in the values we got from parts B and C.
Part E: Write an equation for the energy density due to the electric field between the cylinders in terms of λ, r, and e0.
Starting with the equation for energy density , then substituting the equation for E from part a gives us the equation which simplifies into
Part F: Consider a thin cylindrical shell of thickness dr and radius R1 < r < R2 that is concentric with the cylindrical capacitor. Write an equation for the total energy per unit length contained in the shell in terms of λ, r, dr, and ε0.
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