Question

Rank in order, from most positive to most negative, the potential energies$U_{\mathrm{a}}$to $U_{\mathrm{f}}$of the six electric dipoles in the uniform electric field of FIGURE Q25.5. Explain.

Short answer

Rank of the dipole from most positive to negative is $d,c,e=f,b,a$

Step-by-step solution

Given information and formulae used

Figure of a six-pole dipole in an electric field that is uniform.

Figure.1

Formula used:

The dipole moment's potential energy is calculated using the formula.

$U=-PE\cos \theta$

Where,

$U$is potential energy

$P$is dipole moment

$\mathit{E}$is electric field

$\theta$is the angle of dipole from the horizontal

Sort the dipoles from most positive to least positive. 

(a)

Assuming that first dipole is at $0°$

$\begin{array}{rcl}{U}_a& =& -PE\cos 0\\ & =& -PE\end{array}$

(b)

Assuming that dipole is at$\theta °$

$U_b=-PE\cos \theta$

(c)

The dipole is at $90+\theta °$

$\begin{array}{rcl}{U}_c& =& -PE\cos (90+\theta )\\ & =& +PE\sin \theta \left(\text{here}0°<\theta <90°\right)\end{array}$

(d)

The dipole is at $180°$

$\begin{array}{rcl}{U}_d& =& -PE\cos 180\\ & =& +PE\end{array}$

(e)

The dipole is at $90°$

$\begin{array}{rcl}{U}_e& =& -PE\cos 90\\ & =& 0\end{array}$

(f)

The dipole is at $270°$

$\begin{array}{rcl}{U}_f& =& -PE\cos 270\\ & =& 0\end{array}$

Thus, the order of potential energy increases. $d,c,e=f,b,a$.