2.5 Find the electric field a distance \(z\) above the centre of a circular loop of radius \(r\), which carries a uniform line charge \(\lambda\).

$$d \vec{E} = \frac{ 1 }{4 \pi \epsilon_0} \frac{\lambda R}{ ( R^2 + z^2 )^{3/2}} \left( R \cos \theta \, \hat{i} + R \sin \theta \, \hat{j} + z \, \hat{k} \right) \, d\theta$$

$$\vec{E} = \frac{ 1 }{4 \pi \epsilon_0} \frac{\lambda R}{ ( R^2 + z^2 )^{3/2}} \int_0^{2 \pi} \left( R \cos \theta \, \hat{i} + R \sin \theta \, \hat{j} + z \, \hat{k} \right) \, d\theta$$

$$\vec{E} = \frac{ 1 }{4 \pi \epsilon_0} \frac{2 \pi \lambda R z}{ ( R^2 + z^2 )^{3/2}} \, \hat{k} = \frac{ 1 }{2 \epsilon_0} \frac{\lambda R z}{ ( R^2 + z^2 )^{3/2}} \, \hat{k}$$