Question

What is the electric field (given as a vector) at coordinates (x,y,z) = (1m,2m,2m) produced by a point charge of 4 μC sitting at the origin.

Answer 1

`\vec{E} = 1.334\hat{i} + 2.668\hat{j} + 2.668\hat{k}` N/C

Solution:

Charge, Q =
`4\mu C = 4* 10^(- 6) C`

Coordinates (x, y, z) are (1, 2, 2)

Now, we know that the Electric field at any point in the Cartesian plane is given by:

`\vec{E} = (1)/(4\pi epsilon_(o))* (Q)/(|d|^(2))\hat{d}` (1)

where

`(1)/(4\pi epsilon_(o)) = 9* 10^(9) N-m^(2)/C^(2)`

d = distance of the charge from the point

`\hat{d} = \hat{i} + 2\hat{j} + 2\hat{k}`

`|d| = \sqrt{1^(2) + 2^(2) + 2^(2)`

Now, from eqn (1):

`\vec{E} = 9* 10^(9)* \frac{4* 10^(- 6)}{(\sqrt{1^(2) + 2^(2) + 2^(2))^(2)}}* (\hat{i} + 2\hat{j} + 2\hat{k})`

`\vec{E} = 1.334(\hat{i} + 2\hat{j} + 2\hat{k})` N/C

`\vec{E} = 1.334\hat{i} + 2.668\hat{j} + 2.668\hat{k}` N/C