Acceleration of Point Charge Due to Two Other Point Charges

Question

Example
A negative test charge sits 1.4 m away from the middle point between a pair of positive charges. The positive charges are separated by 1.0 m and each have a charge of 2.0 mc.
What is the electric field vector at the position of the negative test charge?

Answer 1

To find the electric field vector at the position of the negative test charge, we need to calculate the electric field contributions from each of the positive charges and then combine them vectorially.

The electric field due to a point charge is given by the equation:

E = k * (q / r^2) * r̂

where E is the electric field vector, k is the electrostatic constant (9 x 10^9 N m^2/C^2), q is the charge of the point charge, r is the distance from the point charge to the location where the electric field is being measured, and r̂ is a unit vector pointing from the point charge to the location.

Let's call the first positive charge Q1 and the second positive charge Q2. The electric field vector due to each positive charge at the position of the negative test charge can be calculated as follows:

Electric field due to Q1:
E1 = k * (Q1 / r1^2) * r̂

where r1 is the distance from Q1 to the position of the negative test charge (0.7 m, since it is halfway between the positive charges).

Electric field due to Q2:
E2 = k * (Q2 / r2^2) * r̂

where r2 is the distance from Q2 to the position of the negative test charge (0.7 m, since it is halfway between the positive charges).

Now, to find the total electric field vector at the position of the negative test charge, we need to add the electric field vectors due to each positive charge:

E_total = E1 + E2

Since the electric field vectors are vectors, we need to consider their directions as well. The electric field vectors from each positive charge will point radially away from that charge. In this case, since both positive charges have the same magnitude, their electric field vectors will have the same magnitude but opposite directions, effectively cancelling each other out. Therefore, the total electric field vector at the position of the negative test charge will be zero.