How to Solve a Coulomb's Law Problem with Multiple Charges (in 1D & Charges on Angles)

This video will cover Coulomb's Law problems involving multiple charges, both in one dimension and two dimensions (with forces at angles). It assumes familiarity with basic direct application of Coulomb's Law to find electrostatic forces.

For a one-dimensional problem with three charges (q1, q2, q3) in a row, to find the net force on q2: calculate the force between q1 and q2, and the force between q3 and q2 separately using Coulomb's Law. Determine the direction of these forces (attraction or repulsion) based on charge signs. Then, add or subtract the forces based on their directions to find the net force. For example, if both forces push/pull in the same direction, add them; if in opposite directions, subtract them.

A more complex problem involves charges in two dimensions, where finding the net force on q1 requires more steps. First, calculate the individual forces. For instance, the force between q1 and q2, where q1 is positive and q2 is negative, will result in attraction. For the force between q1 and q3, if both are positive, it will be repulsion. The separation distance for the force between q1 and q3 might require using the Pythagorean theorem if laid out as a triangle structure.

The force from q2 on q1 (attraction) pulls q1 downwards. The force from q3 on q1 (repulsion) pushes q1 away at an angle. Since one force is at an angle, it must be broken down into its horizontal and vertical components using trigonometry (sine and cosine). A 45-degree angle simplifies this, as both components will be equal.

Combine all vertical components and all horizontal components. In this example, there are two vertical forces (one pulling down, one pushing up due to the angled force components) which are subtracted to find the net vertical force. The horizontal force component remains as is. Finally, use the Pythagorean theorem with the net vertical and horizontal forces to find the magnitude of the overall net force acting on q1.

To find the direction of the net force, use an inverse trigonometric function, such as inverse tangent, with the net vertical and horizontal force components. This yields the angle at which the net force is acting, relative to the horizontal or vertical axis.

The overall process involves: 1) applying Coulomb's Law for each pair of charges, 2) drawing vectors to visualize force directions, 3) for angled forces, breaking them into horizontal and vertical components, and 4) combining all components to find the net force magnitude and direction.