Acceleration of Point Charge Due to Two Other Point Charges
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?
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.
To find the electric field vector at the position of the negative test charge, we will use Coulomb's Law. Coulomb's Law states that the electric force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Electric Field (E) = Force (F) / Test Charge (q)
1. Calculate the electric force on the negative test charge due to the first positive charge:
We can use Coulomb's Law to find the electric force (F1) between the first positive charge and the test charge.
F1 = k * (q1 * q) / r1^2
Where:
- k is the electrostatic constant, approximately equal to 9.0 x 10^9 N·m^2/C^2.
- q1 is the charge of the first positive charge, which is 2.0 mc.
- q is the charge of the negative test charge.
- r1 is the distance between the first positive charge and the test charge, which is 1.4 m.
2. Calculate the electric force on the negative test charge due to the second positive charge:
We can use Coulomb's Law to find the electric force (F2) between the second positive charge and the test charge.
F2 = k * (q2 * q) / r2^2
Where:
- q2 is the charge of the second positive charge, which is 2.0 mc.
- r2 is the distance between the second positive charge and the test charge. In this case, since the test charge is located at the midpoint between the two positive charges, r2 is equal to half of their separation distance, which is 0.5 m.
3. Calculate the net electric force on the negative test charge:
The net electric force on the negative test charge will be the vector sum of the forces due to the two positive charges.
Fnet = F1 + F2
4. Calculate the electric field at the position of the negative test charge:
Now, we can find the electric field (E) at the position of the negative test charge using the net electric force and the charge of the test charge.
E = Fnet / q
Substituting the values into the equations will give you the final answer for the electric field vector at the position of the negative test charge.
To determine the electric field vector at the position of the negative test charge, we need to first calculate the electric field due to each of the positive charges individually. Then, we can calculate the net electric field vector by summing the individual electric fields.
Here's how to do it step-by-step:
1. Calculate the electric field due to each positive charge using Coulomb's law. The electric field due to a point charge is given by the equation:
E = k * (q / r^2)
where E is the electric field, k is the electrostatic constant (9.0 x 10^9 Nm^2/C^2), q is the charge of the point charge, and r is the distance between the point charge and the location where we want to calculate the electric field.
For each of the positive charges, q = 2.0 mc (microcoulombs) and r = 1.4 m (distance to the middle point between the charges).
For the left positive charge, we have:
E1 = k * (q / r^2)
For the right positive charge, we have:
E2 = k * (q / r^2)
2. Calculate the net electric field vector by adding the individual electric fields together. Since the electric fields due to the positive charges have the same direction (away from the charges), we can simply add them:
E_net = E1 + E2
This will give you the net electric field vector at the position of the negative test charge. Note that the electric field vector is a vector quantity, meaning it has both magnitude and direction.