Question

A dipole is located at the origin, and is composed of charged particles with charge +e and -e, separated by a distance 2x10^-10 along the axis. (a) Calculate the magnitude of the electric field due to this dipole at location $<0,2\times {10}^{-8},0>\mathit{m}$. (b) Calculate the magnitude of the electric field due to this dipole at location $<2\times {10}^{-8},0,0>\mathit{m}$.

Short answer

1. The magnitude of the electric field due to this dipole at locationis $3.6\times {10}^{4}N/C$.
2. The magnitude of the electric field due to this dipole at locationis $3.6\times {10}^{4}N/C$.

Step-by-step solution

Identification of the given data

The given data can be listed below as:

• The magnitude of the electric charge that forms a dipole is $e=1.6\times {10}^{-19}C$.
• The separation distance between the charges is $s=2\times {10}^{-10}m$.
• The location of the dipole is .
• The location of a point is .
• The location of another point is .

Significance of electric dipole

Whenever two electric charges that consist of different natures lie at a specific distance, then both the electric charges form a dipole that refers to an electric dipole. The electric dipole has a direct linear relationship with the magnitude of the electric charge.

(a) Determination of the magnitude of the electric field due to this dipole at location <0,2×10-8,0>m.

The expression of the position vector from the origin to the specified location can be expressed as:

Here, $\stackrel{\rightharpoonup }{r}$represents the position vector from the origin to the specified location.

Substitute all the values in the above equation.

The magnitude of the distance from the origin to the specified location can be calculated as:

$\begin{array}{rcl}r& =& \left|\stackrel{\rightharpoonup }{r}\right|\\ & =& \sqrt{{\left(0m\right)}^2+{\left(2\times {10}^{-8}m\right)}^2+{\left(0m\right)}^2}\\ & =& 2\times {10}^{-8}m\end{array}$

The expression of the magnitude of the electric field due to this dipole at location role="math" localid="1661163673926" can be expressed as:

$E=k\left(\frac{es}{r^3}\right)$

Here, E represents the magnitude of the electric field due to this dipole at location and k represents the Coulomb’s constant whose value is $9\times {10}^{9}N.m^2/C^2$.

Substitute the values in the above equation.

$\begin{array}{rcl}E& =& \left(9\times {10}^{9}N.m^2/C^2\right)\left[\frac{\left(1.6\times {10}^{-19}C\right)\left(2\times {10}^{-10}m\right)}{\left(2\times {10}^{-8}m\right)}\right]\\ & =& 3.6\times {10}^{4}N/C\end{array}$

Hence, the magnitude of the electric field due to this dipole at location is $3.6\times {10}^{4}N/C$.

(b) Determination of the magnitude of the electric field due to this dipole at location <2×10-8,0,0>m.

The expression of the position vector from the origin to the specified location can be expressed as:

Here, $\stackrel{\rightharpoonup }{r}$represents the position vector from the origin to the specified location.

Substitute all the values in the above equation.

The magnitude of the distance from the origin to the specified location can be calculated as:

$\begin{array}{rcl}r& =& \left|\stackrel{\rightharpoonup }{r}\right|\\ & =& \sqrt{{\left(2\times {10}^{-8}m\right)}^2+{\left(0m\right)}^2+{\left(0m\right)}^2}\\ & =& 2\times {10}^{-8}m\end{array}$

The expression of the magnitude of the electric field due to this dipole at location can be expressed as:

$E=k\left(\frac{es}{r^3}\right)$

Here, E represents the magnitude of the electric field due to this dipole at location role="math" localid="1661165332251" and k represents the Coulomb’s constant whose value is $9\times {10}^{9}N.m^2/C^2$.

Substitute the values in the above equation.

$\begin{array}{rcl}E& =& \left(9\times {10}^{9}N.m^2/C^2\right)\left[\frac{\left(1.6\times {10}^{-19}C\right)\left(2\times \times {10}^{-10}m\right)}{{\left(2\times {10}^{-8}m\right)}^3}\right]\\ & =& 3.6\times {10}^{4}N/C\end{array}$

Hence, the magnitude of the electric field due to this dipole at location is $\begin{array}{rcl}& & 3.6\times {10}^{4}N/C\end{array}$.