A long coaxial cable consists of an inner cylindrical conductor with radius a and an outer coaxial cylinder with inner radius b and outer radius c. The outer cylinder is mounted on insulating supports and has no net charge. The inner cylinder has a uniform positive charge per unit length λCalculate the electric field
(a) At any point between the cylinders a distance r from the axis and
(b) At any point outside the outer cylinder.
(c) Graph the magnitude of the electric field as a function of the distance r from the axis of the cable, from r = 0 to r = 2c.
(d) Find the charge per unit length on the inner surface and on the outer surface of the outer cylinder.

Answer:

Part a)

Part b)

Part d)

As we know that due to induction of charge there will be same charge appear on the inner and outer surface of the cylinder but the sign of the charge must be different

On the inner side of the cylinder there will be negative charge induce on the inner surface and on the outer surface of the cylinder there will be same magnitude charge with positive sign.

Explanation:

Part a)

By Guass law we know that

Part b)

Outside the outer cylinder we will again use Guass law

Part d)

As we know that due to induction of charge there will be same charge appear on the inner and outer surface of the cylinder but the sign of the charge must be different

On the inner side of the cylinder there will be negative charge induce on the inner surface and on the outer surface of the cylinder there will be same magnitude charge with positive sign.

Final answer:

The electric field between the cylinders is given by E = λ / (2πε₀r). The electric field outside the outer cylinder is zero due to the absence of net charge. Graph the electric field magnitude using the equation E = λ / (2πε₀r). The inner surface charge of the outer cylinder is -λ and the outer surface charge is 0.

Explanation:

To calculate the electric field between the cylinders at a distance r from the axis, you can use Gauss's Law. Since the charging is uniform, the electric field will also be uniform. Therefore, the electric field at any point between the cylinders is given by E = λ / (2πε₀r), where ε₀ is the permittivity of free space.

To calculate the electric field at any point outside the outer cylinder, you can use the principle of superposition. The electric field due to the outer cylinder is zero because it has no net charge. The electric field due to the inner cylinder can be calculated using the same formula as before.

To graph the magnitude of the electric field as a function of the distance r from the axis, you can plot the equation E = λ / (2πε₀r) for values of r ranging from 0 to 2c.

The charge per unit length on the inner surface of the outer cylinder is -λ, while the charge per unit length on the outer surface of the outer cylinder is 0. This is because the outer cylinder has no net charge and the inner cylinder has a uniform positive charge per unit length λ.

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