A uniformly charged rod of length L carries a linear charge density $\lambda$ and lies along the x‑axis centred at the origin (from $-L/2$ to $+L/2$). A point P is located a distance a above the centre of the rod on the perpendicular bisector.

(a) Derive an expression for the magnitude of the electric field at P due to the rod.

(b) A second identical rod is placed parallel to the first at a horizontal separation d (the rods are coplanar). Find the net electric field magnitude at the midpoint between the two rods.

(c) If a uniform external electric field $E_{\text{ext}}$ is applied opposite to the net field found in part (b), determine the magnitude of $E_{\text{ext}}$ required to make the total electric field at that midpoint zero. Express your answer in terms of $\lambda, L, a$ and $\varepsilon_{0}$.

Question

A uniformly charged rod of length L carries a linear charge density $\lambda$ and lies along the x‑axis centred at the origin (from $-L/2$ to $+L/2$). A point P is located a distance a above the centre of the rod on the perpendicular bisector.

(a) Derive an expression for the magnitude of the electric field at P due to the rod.

(b) A second identical rod is placed parallel to the first at a horizontal separation d (the rods are coplanar). Find the net electric field magnitude at the midpoint between the two rods.

(c) If a uniform external electric field $E_{\text{ext}}$ is applied opposite to the net field found in part (b), determine the magnitude of $E_{\text{ext}}$ required to make the total electric field at that midpoint zero. Express your answer in terms of $\lambda, L, a$ and $\varepsilon_{0}$.

Accepted Answer

E_{ext}=\frac{\lambda L}{2\pi\varepsilon_{0} a\sqrt{a^{2}+(L/2)^{2}}}

Gce Model Questions Vol1 2022 Physics Question 27

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