Question

What is the strength of an electric field resulting from a dipole with a dipole moment of 25 C⋅m at a radius, r = 30 cm away and 75° from the y-axis?

Answer 1

Let the direction of dipole is along the positive direction of x-axis.

Since,

$${\displaystyle \mathbf {E} \left(\mathbf {r} \right)={\frac {3\left(\mathbf {p} \cdot {\hat {\mathbf {r} }}\right){\hat {\mathbf {r} }}-\mathbf {p} }{4\pi \varepsilon _{0}r^{3}}}\ .}$$

And,

$$\bold p=25\hat{i}\\ \bold r=0.3(\cos(15^\circ)\hat{i}+\sin(15^\circ)\hat{j})$$

Now.

\bold p\cdot\hat\bold r=25\cos(15^\circ)

Thus,

$$\bold E(\bold r)=\frac{9\times 10^9}{0.3^3}(3(25\cos(15^\circ))(\cos(15^\circ)\hat{i}+\sin(15^\circ)\hat{j})-25\hat{i}\\ \implies \bold E=\frac{10^{12}}{3}(44.97\hat{i}+18.75\hat j)\\ \implies E=16.24\times10^{12}N/C$$