Question

Analysis Model: Particle in a Field (Electric)
Two equal positively charged particles are at opposite corners of a trapezoid as shown in Figure. Find symbolic expressions for the total electric field at (a) the point $$P$$ and (b) the point $$P'$$.

Answer 1

(a) See Figure (a). The distance from the $$+Q$$ charge on the upper left is $$d$$, and the distance from the $$+Q$$ charge on the lower right to point $$P$$ is
$$\sqrt{(d/2)^2+(d/2)^2}$$
The total electric field at point $$P$$ is then
$$\overrightarrow{E}_p=k_e\dfrac{Q}{d^2}\hat{i}+k_e\dfrac{Q}{[(d/2)^2+(d/2)^2]}\left(\dfrac{-\hat{i}+\hat{j}}{\sqrt{2}}\right)$$
$$=k_e\left[\dfrac{Q}{d^2}\hat{i}+\dfrac{Q}{d^2/2}\left(\dfrac{-\hat{i}+\hat{j}}{\sqrt{2}}\right)\right]$$
$$=\boxed{k_e\dfrac{Q}{d^2}\left[(1-\sqrt{2})\hat{i}+\sqrt{2}\hat{j}\right] \,}$$
(b) See Figure (b). The distance from the $$+Q$$ charge on the lower right to point $$P^1$$ is $$2d$$, and the distance from the $$+Q$$ charge on the upper right to point $$P^1$$ is
$$\sqrt{(d/2)^2+(d/2)^2}$$
The total electric field at point $$P^1$$ is then
$$\overrightarrow{E}_{P^1}=k_e\dfrac{Q}{[(d/2)^2+(d/2)^2]}\left(\dfrac{-\hat{i}-\hat{j}}{\sqrt{2}}\right)+k_e\dfrac{Q}{(2d)^2}(\hat{i})$$
$$\overrightarrow{E}_{P^1}=-k_e\left[\dfrac{Q}{d^2/2}\left(\dfrac{-\hat{i}+\hat{j}}{\sqrt{2}}\right)+\dfrac{Q}{4d^2}(\hat{i})\right]$$
$$=-k_e\dfrac{Q}{4d^2}\left[\dfrac{8}{\sqrt{2}}(\hat{i}+\hat{j})+(\hat{i})\right]$$
$$\overrightarrow{E}_{P^1}=\boxed{-k_e\dfrac{Q}{4d^2}\left[(1+4\sqrt{2})\hat{i}+4\sqrt{2}\hat{j}\right] \,}$$