Session no 21 SWAYAM : Examples on electric field density part -1

Main Ideas and Methods

The session demonstrates how to calculate electric-field intensity from point charges in free space using displacement vectors, Coulomb’s law, and superposition.

Example 1: Field at One Charge’s Location Due to Another Charge

A (2\,\mu\text{C}) source charge is at (P_1=(1,0,0)), and the field is required at (P_2=(0,1,0)). A (1\,\mu\text{C}) charge is also specified at (P_2), but it does not affect the field produced there by the source charge.

The displacement from source to destination is

[ \mathbf R=\mathbf r_{P_2}-\mathbf r_{P_1} =-\mathbf a_x+\mathbf a_y. ]

Its magnitude is (\sqrt{2}), so the direction unit vector is

[ \mathbf a_R=\frac{-\mathbf a_x+\mathbf a_y}{\sqrt{2}}. ]

Using the point-charge field formula,

[ \mathbf E=\frac{q}{4\pi\epsilon_0R^2}\mathbf a_R, ]

the resulting field is approximately

[ \mathbf E=6.364(-\mathbf a_x+\mathbf a_y)\ \text{kN/C}. ]

Example 2: Total Field from Several Charges

The electric-field vectors at point (P) are

[ \mathbf E_1=\mathbf a_x+2\mathbf a_y-\mathbf a_z, \quad \mathbf E_2=\mathbf a_y+3\mathbf a_z, \quad \mathbf E_3=2\mathbf a_x-\mathbf a_y ]

in N/C.

By the superposition principle, add the field vectors component by component:

[ \mathbf E_{\text{total}} =\mathbf E_1+\mathbf E_2+\mathbf E_3 =3\mathbf a_x+2\mathbf a_y+2\mathbf a_z\ \text{N/C}. ]

Example 3: Choose a Charge So the Field’s (x)-Component Is Zero

A (-10\text{ nC}) charge is at the origin. The field is considered at (P=(3,1,1)), and an unknown charge (Q_2) is placed at ((2,0,0)).

The displacement vectors from each source to (P) are

[ \mathbf R_1=(3,1,1),\quad |\mathbf R_1|=\sqrt{11}; \qquad \mathbf R_2=(1,1,1),\quad |\mathbf R_2|=\sqrt{3}. ]

Calculate each charge’s field using

[ \mathbf E=\frac{q}{4\pi\epsilon_0R^2}\mathbf a_R. ]

Set the sum of the two fields’ (x)-components to zero and solve for the unknown charge. The required charge is approximately

[ Q_2=4.27\text{ nC}. ]

General Method

Identify the source and destination, subtract the source coordinates from the destination coordinates to find the displacement vector, calculate its magnitude and unit vector, and apply Coulomb’s law. When multiple charges are involved, add their field contributions.

Speakers and Sources

• Speaker: One lecturer/presenter; the name is not given in the subtitles.
• Source/channel: Mother India Today.