Question:

A point charge of +4 μC is located at the origin in the xy-plane. A second point charge of -2 μC is located at (3,4) on the xy-plane.

a) Calculate the electric field due to the +4 μC charge at the point (1,2). b) Determine the total electric field at the point (3,4) due to both charges.

Answer:

a) The electric field due to a point charge can be calculated using the formula:

where $k = 8.99 \times 10^9 \, Nm^2/C^2$ (Coulomb's constant), $q$ is the magnitude of the point charge, $r$ is the distance from the point charge, and $\hat{r}$ is the unit vector pointing from the charge to the point of interest.

Given that $q = +4 \, \mu C = 4 \times 10^{-6} \, C$ and the distance $r = \sqrt{(1-0)^2 + (2-0)^2} = \sqrt{5}$.

Plugging these values into the formula, [ \vec{E}_{+4\mu C} = \frac{(8.99 \times 10^9 , Nm^2/C^2) \cdot (4 \times 10^{-6} , C)}{5} \cdot \hat{r} ]

The unit vector $\hat{r}$ for the point (1,2) is $\frac{1}{\sqrt{5}} \vec{i} + \frac{2}{\sqrt{5}} \vec{j}$. Therefore, [ \vec{E}_{+4\mu C} = (7.19 \times 10^6 , N/C) \cdot \left( \frac{1}{\sqrt{5}} \vec{i} + \frac{2}{\sqrt{5}} \vec{j} \right) ]

b) The total electric field at the point (3,4) due to both charges can be calculated by summing the electric fields due to each individual charge.

For the +4 μC charge at the origin, the electric field at (3,4) would be: [ \vec{E}_{+4\mu C} = \frac{(8.99 \times 10^9 , Nm^2/C^2) \cdot (4 \times 10^{-6} , C)}{5^2} \cdot \hat{r} = (1.44 \times 10^6 , N/C) \cdot \hat{r} ]

The unit vector $\hat{r}$ for the point (3,4) is $\frac{3}{5} \vec{i} + \frac{4}{5} \vec{j}$. Therefore, [ \vec{E}_{+4\mu C} = (1.44 \times 10^6 , N/C) \cdot \left( \frac{3}{5} \vec{i} + \frac{4}{5} \vec{j} \right) ]

For the -2 μC charge at (3,4) the electric field magnitude and unit vector are: [ \vec{E}_{-2\mu C} = \frac{(8.99 \times 10^9 , Nm^2/C^2) \cdot (2 \times 10^{-6} , C)}{0^2} \cdot \hat{r} = \infty ]

Since the point is on the charge itself, the electric field due to the -2 μC charge at (3,4) is infinite due to the singularity at the location of the charge.

Adding the electric fields due to both charges gives the total electric field at (3,4): [ \vec{E}{total} = \vec{E}{+4\mu C} + \vec{E}_{-2\mu C} = (1.44 \times 10^6 , N/C) \cdot \left( \frac{3}{5} \vec{i} + \frac{4}{5} \vec{j} \right) + \infty ]

Thus, the total electric field at the point (3,4) is undefined due to the singularity at the location of the -2 μC charge.