A ring of radius R has uniform linear charge density lamda the electric field at the centre of ring is?

Question

Answer

Electric Field at the Centre of a Ring of Uniform Charge Density

When we have a ring of radius R with uniform linear charge density lamda, we can find the electric field at the centre of the ring using the following steps:

Find the electric field due to a small element of charge

We can start by considering a small element of charge on the ring. This element has charge dq = lamda dl, where dl is the length of the element. The electric field due to this element at the centre of the ring is given by:
dE = kdq/r^2
where k is the Coulomb constant and r is the distance between the element and the centre of the ring. In this case, r is simply equal to R, since the element is on the ring itself.

Find the total electric field

To find the total electric field at the centre of the ring, we need to consider the contribution from all the elements of charge on the ring. Since the ring has a uniform charge density, we can divide it into small elements of charge, each of length dl. The total electric field at the centre of the ring is then given by integrating over all the elements:
E = ∫dE = k∫(lamda dl)/R^2

Simplify the integral

We can simplify the integral by noting that the total charge on the ring is given by Q = lamda L, where L is the circumference of the ring (2πR). We can therefore express lamda as Q/L, and the integral becomes:
E = kQ/R^2 ∫(1/L)dl = kQ/R^2 (L/L) = kQ/R^2

Final result

We have now found the electric field at the centre of the ring:
E = kQ/R^2
This result shows that the electric field at the centre of a ring of uniform charge density is independent of the radius of the ring and only depends on the total charge on the ring. It also shows that the electric field at the centre of the ring is directed along the axis of the ring.

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