A charge \(Q\) is spread uniformly round a thin wire bent into a circle O radius a. \(\mathrm{P}\) is a point on its axis at a distance \(r\) from the wire itself and \(\theta\) is the angle between \(r\) and the axis. Show that the E-field at P has a magnitude \(Q \cos \theta /\left(4 \pi e_{0} r^{2}\right)\). Show from that result that the work done in taking unit charge from \(O\) to infinity is \(Q /\left(4 \pi \epsilon_{0} a\right)\),

The electric field intensity is a measure of the force per unit charge exerted at any point in space due to electric charges. Understanding its intensity is crucial to grasping electric phenomena.
In the problem provided, we consider a charge, Q, that is uniformly distributed around a circular wire, which creates an electric field in the surrounding space. The intensity of the field at a point P is found using the principle that the electric field due to a point charge decreases with the square of the distance from the charge, which follows Coulomb's law.
A key step in solving for the field intensity involves integrating the effects of infinitesimal charge elements over the entire wire, taking into account that only components parallel to the defined axis will contribute. As a result, the magnitude of the electric field at P was shown to be Q cos(θ)/(4 π ε0 r²), indicative of an inverse square law relationship but modified by the cosine of the angle, reflecting the directional nature of electric field components.

Uniform charge distribution is a term that refers to an even spread of electric charge over a surface or length, resulting in a constant charge density. This concept is essential in simplifying calculations involving the electric field and potential.
In our circle wire example, the charge, Q, distributes uniformly around the wire's circumferece, which implies that any segment of the wire will contain a charge proportional to its length. This fact enables us to express the infinitesimal charge element dq as Q/(2πa) dl, where dl is a small segment of the wire's circumference.
Understanding this uniformity is critical because it ensures that the perpendicular components of the electric field from different parts of the wire will cancel out, while the parallel components will add up coherently. This leads to a simplified expression for the total electric field in terms of a single integral over the angle θ, avoiding the complexity of a more involved vector sum.

Work done by the electric field on a charge involves moving the charge within the field and is given by the integral of the force along the path taken. It's a fundamental concept linking electric fields to energy and potential.
In our discussion of moving a unit charge from the center of the circular wire to infinity, the work done is obtained by integrating the electric field over the distance from the wire to an infinitely distant point. By considering a path along the axis and factoring in that the electric field strength diminishes as the inverse square of the distance, the integration yields a result that represents the potential difference between the wire and a point at infinity.
This potential difference, when multiplied by the charge amount, gives the total work done, which in the case of a unit charge reflects the electric potential due to the wire. The conclusion that the work done is Q/(4 π ε0 a) supports the notion that potential at a surface depends inversely on the distance from the charge distribution, a cornerstone of electrostatics.