Using radially outward as the positive +r direction, two long, charged, thin-wal

Question

Using radially outward as the positive +r direction, two long, charged, thin-walled, concentric cylinders have radii of a = 3.0 cm and b = 6.0 cm. The charge per unit length is A = 5.3x10-6 C/m on the inner shell and B = -9.0x10-6 C/m on the outer shell.

1. Determine the magnitude and direction of the electric field at a radial distance of r = 4.2 cm from the common central axis.

2. Determine the magnitude and direction of the electric field at a radial distance of r = 8.3 cm from the common central axis.

Explanation / Answer

The E-field will be radial by symmetry and therefore normal to any concentric cylindrical surface.

So take a Gaussian surface of radius r and length L centered on the central axis. Then Gauss theorem gives

2pirLE(r) = Qenclosed/0

2pi 0rLE(r) = Qenclosed

2pi 0rLE(r) = L

E(r) =net/2pi0r

Thus,

1. E(r) =1/2pi0r if a<r<b

Plugging values,

E(r) = (5.3*10^-6)/(2*3.14*8.85*10^-12*0.042) = 2.27*10^6 N/C

2. E(r) =1/2pi0r if a<r<b

Plugging values,

E(r) = (-9.0*10^-6)/(2*3.14*8.85*10^-12*0.083) = -195*10^6 N/C

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