Questions: Part A The electric field midway between two equal but opposite point charges is 588 N / C, and the distance between the charges is 16.0 cm. What is the magnitude of the charge on each?

Solution Steps

We are given two equal but opposite point charges and the electric field at the midpoint between them. The distance between the charges is 16.0 cm, and the electric field at the midpoint is \(588 \, \mathrm{N/C}\). We need to find the magnitude of each charge.

The electric field \(E\) due to a point charge \(q\) at a distance \(r\) is given by:

\[ E = \frac{k \cdot |q|}{r^2} \]

where \(k = 8.99 \times 10^9 \, \mathrm{N \cdot m^2/C^2}\) is Coulomb's constant.

The total distance between the charges is 16.0 cm, so the distance from each charge to the midpoint is half of that:

\[ r = \frac{16.0 \, \mathrm{cm}}{2} = 8.0 \, \mathrm{cm} = 0.080 \, \mathrm{m} \]

Since the charges are equal and opposite, the electric fields due to each charge at the midpoint add up. The total electric field at the midpoint is given by:

\[ E = \frac{2k \cdot |q|}{r^2} \]

Given \(E = 588 \, \mathrm{N/C}\), we can substitute the known values:

\[ 588 = \frac{2 \cdot 8.99 \times 10^9 \cdot |q|}{(0.080)^2} \]

Rearrange the equation to solve for \(|q|\):

\[ |q| = \frac{588 \cdot (0.080)^2}{2 \cdot 8.99 \times 10^9} \]

Calculate \(|q|\):

\[ |q| = \frac{588 \cdot 0.0064}{17.98 \times 10^9} \]

\[ |q| = \frac{3.7632}{17.98 \times 10^9} \]

\[ |q| = 2.092 \times 10^{-10} \, \mathrm{C} \]

Final Answer