Q58P - Fundamentals Of Physics

Question

A certain electric dipole is placed in a uniform electric field  $\overrightarrow{\mathbf{E}}$ of magnitude.$\mathbf{20}\mathbf{ }\mathbf{\text{N/C}}$ Figure 22-62 gives the potential energy of the dipole versus the angle u between E and the dipole moment  $\overrightarrow{\mathbf{p}}$. The vertical axis scale is set by${\mathit{U}}_{\mathbf{s}}\mathbf{=}\mathbf{ }\mathbf{1}\mathbf{.00}\mathbf{ }\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{28}}\mathbf{\text{J}}$ .What is the magnitude of $\overrightarrow{\mathbf{p}}$ ?

 

Step-by-Step Solution

Answer

The magnitude of dipole moment $\overrightarrow{p}$ is.$5.00\times {10}^{-28}\text{C.m}$

Step 1: The given data

1. Electric field of magnitude,$E=20\text{ N/C}$
2. And the above graph,$U_s=1.00\times {10}^{-28}\text{J}$

Step 2: Understanding the concept of energy of the electric dipole

The potential energy of the electric dipole placed in an electric field depends on its orientation relative to the electric field.The magnitude of the electric dipole moment is$p=qd$, where q is the magnitude of the charge, and d is the separation between the two charges.

 

Formula:

When placed in an electric field, the potential energy of the dipole,

                                                                                   (i)

 $\begin{array}{c}U\left(\theta \right)=-\overrightarrow{p}.\overrightarrow{E}\\ =-pE\cos \theta \end{array}$

Step 3: Calculation of magnitude of the dipole moment

 

Examining the lowest value on the graph, we have the magnitude of the dipole moment using equation (i) as:

$\begin{array}{c}U=-\overrightarrow{p}.\overrightarrow{E}\\ - 1.00\times {10}^{-28}\text{J}=-20\text{ N/C}\left(\overrightarrow{p}\right)\\ \overrightarrow{p}=5.00\times {10}^{-28}\text{C.m}\end{array}$

Hence, the value of the dipole moment is.$5.00\times {10}^{-28}\text{C.m}$