Question

A dipole is placed parallel to the electric field. If W is the work done in rotating the dipole by ${60}^o$, then the work done in rotating it by 180º is

Answer 1

Step1: Given data

We have given that initially the dipole is placed parallel to the electric field. This means that the initial angle between the dipole and electric field is $0^o$. ${\theta }_1=0^o$∘

Let the work done in rotating the dipole from the initial parallel position to ${\theta }_2={60}^o$ is W. ∘

Step2: Formula used

The work done in rotating the dipole placed in the electric field is $W=PE(\cos {\theta }_1-\cos {\theta }_2)................\left(i\right)$

Here, $P$ is the dipole moment, $E$ is the electric field, ${\theta }_1$ is the initial angle of the dipole with the electric field and ${\theta }_2$ is the final angle of the dipole moment with the electric field.

Step3: Calculating work done in rotating the dipole from the initial parallel position to ${60}^o$

The work done in rotating the dipole from the initial parallel position to ${60}^o$ is given by equation (1). Substitute ${\theta }_1=0^o$ and ${\theta }_2={60}^o$ in equation (1).

$$\begin{gathered} W=PE(\cos {\theta }_1-\cos {\theta }_2) \\ W==PE(\cos 0^o-\cos {60}^o) \\ W=PE\left(1-\frac{1}{2}\right)[As\cos {60}^o=\frac{1}{2}] \\ W=\frac{PE}{2} \\ PE=2W \end{gathered}$$

Step4: Calculating the work done in rotating it by ${180}^o$

Let $W\text{'}$ be the work done in rotating the dipole from $0^o$ to ${180}^o$. Here, the initial angle between the dipole and electric field is $0^o$ and the final angle between the dipole and electric field is ${180}^o$. ${\theta }_1=0^o$. Substitute $W\text{'}$ for $W$, ${\theta }_1=0^o$ and ${\theta }_2={180}^o$ in equation (1).

$$\begin{gathered} W\text{'}=PE(\cos {\theta }_1-\cos {\theta }_2) \\ W\text{'}=PE(\cos 0^o-\cos {180}^o) \\ W\text{'}=PE\left(1-(-1)\right) \\ W\text{'}=2PE \end{gathered}$$

Substituting the value of $PE$

$W\text{'}=2\left(2W\right)=4W$.

Hence, Therefore, the work done in rotating the dipole to ${180}^o$ is $4W$.