A certain coaxial cable consists of a copper wire, radius a , surrounded by a concentric copper tube of inner radius c . The space between is partially filled (from b out to c ) with material of dielectric constant K. The goal of this problem is to find the capacitance per unit length of this cable. You may neglect edge effects.
Note that for technical reasons, we use the symbol l for charge per unit length, rather than the more typical λ . Do not get confused, l is not a length!
Calculate the radial component of the electric field in the region a<r<b . Express your answer in terms of a , b , c , K , l , r , and epsilon_0
Calculate the radial component of the electric field in the region b<r<c . Express your answer in terms of a , b , c , K , l , r , and epsilon_0
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(b) What is the potential difference V(b)-V(a) (be careful about sign)? Express your answer in terms of a , b ,c ,K , l and epsilon_0
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What is the potential difference V(c)-V(b) (be careful about sign)? Express your answer in terms of a ,b , c , K , l and epsilon_0
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What is the magnitude of the potential difference |V(c)-V(a)| ? Express your answer in terms of a , b , c , K ,l and epsilon_0
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(c) What is the capacitance per unit length? Express your answer in terms of a, b , c , K , l and epsilon_0
To find the capacitance per unit length, we can use the formula:
C = (2πε₀) / ln(b/a)
Where:
C is the capacitance per unit length,
ε₀ is the permittivity of free space (approximately 8.854 × 10⁻¹² F/m),
a is the radius of the inner copper wire,
b is the radius where the dielectric material starts,
ln denotes the natural logarithm function.
So, to calculate the capacitance per unit length, plug in the given values for a, b, and ε₀ into the formula and solve for C.