Question 27: Electrostats & Capacitance

Question

Electric field at point $ (30,30,0) $ due to a charge of $ 0.008 \mu \mathrm{C} $ placed at origin will be, (coordinates are in cm )

Accepted Answer

The electric field $\mathbf{E}$ at point $(30, 30, 0)$ due to a charge of $0.008 \, \mu \mathrm{C}$ placed at the origin can be calculated using the formula:
$  \mathbf{E} = \frac{K q}{r^3} \cdot \mathbf{r}  $
Step 1: Calculate the distance $r$:
$  r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}  $
Substitute the given coordinates:
$ r = \sqrt{30^2 + 30^2 + 0^2} = 30 \sqrt{2} \, 	ext{cm} = 30 \sqrt{2} 	imes 10^{-2} \, 	ext{m}  $
Step 2: Convert charge $q$ to Coulombs:
$  q = 8 	imes 10^{-3} \, \mu\mathrm{C} = 8 	imes 10^{-3} 	imes 10^{-6} \, 	ext{C}  $
Step 3: The position vector $\mathbf{r}$ is:
$ \mathbf{r} = (30 \hat{\mathbf{i}} + 30 \hat{\mathbf{j}}) 	imes 10^{-2} \, 	ext{m}  $
Step 4: Substitute into the electric field formula:
$ \mathbf{E} = \frac{9 	imes 10^9 	imes 8 	imes 10^{-3} 	imes 10^{-6}}{\left(30 \sqrt{2} 	imes 10^{-2}ight)^3} 	imes (30 \hat{\mathbf{i}} + 30 \hat{\mathbf{j}}) 	imes 10^{-2}  $
Calculation:
$ \mathbf{E} = \frac{9 	imes 8 	imes 10^9 	imes 10^{-11}}{27 	imes 2 \sqrt{2} 	imes 10^{-6} 	imes 10^3} 	imes 30(\hat{\mathbf{i}} + \hat{\mathbf{j}}) $
Simplify the expression:
$ \mathbf{E} = \frac{9 	imes 8 	imes 10^2 	imes 3}{27 	imes 2 \sqrt{2}}(\hat{\mathbf{i}} + \hat{\mathbf{j}}) $
$ \mathbf{E} = 200 \sqrt{2}(\hat{\mathbf{i}} + \hat{\mathbf{j}}) \, 	ext{NC}^{-1}  $
Thus, the electric field at the given point is $200 \sqrt{2}(\hat{\mathbf{i}} + \hat{\mathbf{j}}) \, \mathrm{NC}^{-1}$.

Answer

$ 8000 \mathrm{NC}^{-1} $

Answer

$ 4000(\hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathrm{NC}^{-1} $

Answer

$ 200 \sqrt{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathrm{NC}^{-1} $

Answer

$ 400 \sqrt{2}(\hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathrm{NC}^{-1} $

Electrostats & Capacitance

Electric field at point (30,30,0) due to a charge of 0.008 \mu \mathrm{C} placed at origin will be, (coordinates are in cm )

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