What if the charges are not on the x or y axes?

Use the distance formula to find the straight-line distance $r$ between the charges. Calculate the force magnitude, then use the geometric angle between the charges to resolve the force into x and y components using sine and cosine.

Do I include the signs of the charges in the Coulomb's law equation?

No. Use the absolute values of the charges to find the force magnitude. Apply the signs only to determine whether the force is attractive or repulsive when setting up your vectors.

What if the system is not in a vacuum?

Replace the vacuum permittivity constant in Coulomb's constant with the permittivity of the specific uniform medium.

How to solve a Coulomb's law problem with three charges

To find the net electrostatic force on a point charge due to two other point charges, compute the individual force vectors using Coulomb's law and sum them. This method applies whenever the charges are stationary and can be approximated as point charges.

The setup

Define a Cartesian coordinate system. Identify the target charge and the two source charges. Note their exact coordinates and the signs of their charges. Determine the straight-line distance from each source charge to the target charge.

The steps

Draw a free body diagram for the target charge. 2. Calculate the magnitude of the force from the first source charge using Coulomb's law: $F_1 = k |q_{target} q_1| / r_1^2$ . 3. Determine the direction of this force based on charge signs (like charges repel, opposite charges attract) and resolve it into x and y components. 4. Repeat steps 2 and 3 for the second source charge to find its force components. 5. Sum the x-components and y-components separately to find the net force components. 6. Calculate the final magnitude and direction of the net force vector using the Pythagorean theorem and inverse tangent.

Checking the result

Verify that the final net force vector points in the geometric quadrant predicted by your initial free body diagram. Ensure that all distances were converted to meters and all charges to Coulombs before calculation.

Common errors

Students frequently forget to square the distance in the denominator of Coulomb's law. Another common error is plugging the negative signs of charges into the magnitude equation, which can lead to incorrect vector resolution. Always use absolute values for magnitude and determine direction conceptually.

Worked example

Find the net force on $q_0 = 1.0 \mu C$ at $(0,0)$ due to $q_1 = 2.0 \mu C$ at $(0.3 m, 0)$ and $q_2 = -3.0 \mu C$ at $(0, 0.4 m)$ . Use $k = 8.99 imes 10^9 N m^2 / C^2$ .

The distance to $q_1$ is $r_1 = 0.3 m$ . Magnitude $F_1 = k |q_0 q_1| / r_1^2 = (8.99 imes 10^9)(1.0 imes 10^{-6})(2.0 imes 10^{-6}) / (0.3)^2 = 0.20 N$ . Both charges are positive, so $q_1$ repels $q_0$ in the -x direction. The vector is $\vec{F}_1 = -0.20 \hat{i} N$ . The distance to $q_2$ is $r_2 = 0.4 m$ . Magnitude $F_2 = k |q_0 q_2| / r_2^2 = (8.99 imes 10^9)(1.0 imes 10^{-6})(3.0 imes 10^{-6}) / (0.4)^2 = 0.17 N$ . The charges have opposite signs, so $q_2$ attracts $q_0$ in the +y direction. The vector is $\vec{F}_2 = 0.17 \hat{j} N$ . The net force is $\vec{F}_{net} = -0.20 \hat{i} + 0.17 \hat{j} N$ . The magnitude is $\sqrt{(-0.20)^2 + (0.17)^2} = 0.26 N$ .

FAQ

Run your own problem

Answer form

References: University Physics Volume 2, OpenStax · Fundamentals of Physics, Halliday & Resnick