A circular wire of radius R carries a total charge Q distributed uniformly over its circumference. A small length of wire subtending angle teta at the centre is cut off. Find the electric field at the centre due to the remaining portion

To find the electric field at the center due to the remaining portion, we can use the concept of symmetry. Since the remaining portion of the wire has a uniform charge distribution and is symmetrically distributed around the center, we can consider the electric field contribution from each small segment of the remaining wire by considering their radial components only.

Let's consider a small segment of the wire subtending an angle δθ at the center, with a charge δQ. The length of this segment can be approximated as Rδθ, where R is the radius of the wire.

The electric field due to this small segment at the center can be calculated using Coulomb's law:

dE = k * (δQ) / r^2

Here, dE is the electric field due to the segment, δQ is the charge on the segment, r is the distance from the segment to the center (which is constant since we are at the center), and k is the electrostatic constant.

Now, we need to find δQ in terms of δθ. Since the charge is uniformly distributed over the circumference, δQ can be expressed as:

δQ = (Q / 2π) * δθ

where Q is the total charge on the wire.

Substituting this value of δQ into the equation for dE, we get:

dE = k * (Q / 2π) * δθ / r^2

To find the total electric field at the center, we need to integrate this expression over the entire remaining portion of the wire. The range of integration can be from 0 to (2π - θ), as we have cut off an angle θ from the circumference of the wire.

Now, integrating the equation, we get:

E = ∫ dE = ∫[0,2π-θ] k * (Q / 2π) * δθ / r^2

Simplifying, we have:

E = (kQ / 2πr^2) ∫[0,2π-θ] δθ

The integral of δθ over the specified range is just the difference of the angles:

E = (kQ / 2πr^2) * [2π - θ - 0]

E = (kQ / r^2) * (1 - θ / 2π)

Therefore, the electric field at the center due to the remaining portion is given by:

E = (kQ / r^2) * (1 - θ / 2π)

Where k is the electrostatic constant, Q is the total charge on the wire, r is the radius of the wire, and θ is the angle subtended by the cut-off portion at the center.

Treat the problem as the superposition of a complete loop of wire a nd a short length of negatively charged wire with the same absolute value of charge per length. The field at the center due to the complete loop is zero. Treat the negatively charged segment as a point source.