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SparkimusPrime
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A group of students want to test if Gauss's law works or not in some real-life situations.
A solid conducting cylinder of radius a = 2cm and a cylindrical conducting shell of radius b = 8cm are arranged to concentric (coaxial), and a voltage of V_0 = 12volts is connected to the two conductors. The heights of the cylinders are h = 2cm.
Prediction:
Then they predict that, if Gauss's law is valid, the electric field on a Gaussian cylindrical surface must be uniform. Then the magnitude of the electric field at any point between a and b is:
(for uniform electric fields)
E = V / d = 12volts / 6cm = 200 volts / meter
Then they predict the voltage across the outside shell and a middle point P must be:
(where 'r' is the distance from the center to 'p')
V = E(r - a) = 200 volts / meter (5cm - 2cm) = 6 volts
Then they can measure the voltage across the outside shell and a middle point P. If it is 6 volts then Gauss's law is valid and works with the real life situation.
Identify the deficiencies in this line of reasoning.
First I set about to calculate the electric field:
Solve for Q with the capacitance of two coaxial cylinders:E_cylinder = (2 Ke Q) / r^2
C_cylinders = l / (2*Ke*ln(b/a))
C*V = Q
Q = l*V / (2*Ke*ln(b/a))
substituting:
E_cylinder = (l*V) / (ln(b/a)*r^2)
Where 'r' is a distance measured from the center. Trying several distances (r = 5cm -> E = 69.2 Volts / meter, r = 2cm [on the internal cylinder] -> 432.8 Volts / meter) I discovered that the electric field varies with distance, therefore its not uniform.
The equation:
V = E/d
Cannot be applied to this situation.
This is (guess) because less charge from the power source settles on the inner cylinder because of its relative radius compared to the outer cylinder.
Am I correct?
Peter