Charge A, +1 μC, is positioned at the origin of a coordinate system as shown below. Charge B, −1 μC, is fixed at x = 3 m.
(a) Find the electric field on the x axis at x = 6 cm.
(b) At what point on the x axis is the electric field zero?
To find the electric field at a given point, you can use Coulomb's Law. Coulomb's Law states that the electric field at a point due to a point charge is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance between the point charge and the point where you want to find the electric field.
(a) Let's find the electric field at x = 6 cm on the x-axis due to charge A. Since charge B is fixed at x = 3 m, it does not contribute to the electric field at x = 6 cm. The distance between charge A and the point where we want to find the electric field is 6 cm, which is 0.06 m.
The formula to calculate the electric field due to a point charge is:
Electric field (E) = (k * q) / r^2
where k is the Coulomb's constant (k = 9 × 10^9 Nm^2/C^2), q is the charge, and r is the distance.
Substituting the values into the formula:
E = (9 × 10^9 Nm^2/C^2 * 1 μC) / (0.06 m)^2
Simplifying the equation gives:
E = 1.67 × 10^5 N/C directed towards the positive x-axis.
Therefore, the electric field at x = 6 cm on the x-axis is 1.67 × 10^5 N/C directed towards the positive x-axis.
(b) To find the point on the x-axis where the electric field is zero, we need to consider both the electric fields due to charge A and charge B.
At any point on the x-axis, the electric field due to charge A is directed towards the positive x-axis, and the electric field due to charge B is directed towards the negative x-axis. These fields will cancel each other out at a certain point on the x-axis where the magnitudes of the electric fields are equal.
The electric field due to charge A at x = 0 is 1.67 × 10^5 N/C directed towards the positive x-axis. As we move towards the right on the x-axis, the magnitude of this electric field decreases, but it still points towards the positive x-axis.
The electric field due to charge B at x = 3 m is given by the same formula as before, substituting the charge of B and the distance between B and the point on the x-axis. Let's calculate it:
E = (9 × 10^9 Nm^2/C^2 * 1 μC) / (3 m)^2
E = 1 × 10^5 N/C directed towards the negative x-axis.
So, at x = 3 m, we have an electric field of 1 × 10^5 N/C directed towards the negative x-axis due to charge B.
Since charge B is positioned to the right of charge A, the electric field due to charge B will be greater than the electric field due to charge A at any point on the x-axis to the left of x = 3 m.
Therefore, there is a point to the left of x = 3 m on the x-axis where the electric field is zero. To find this point, you can calculate the electric fields for different distances from charge A until you find a point where the magnitudes of the electric fields from charges A and B are equal.
In this case, you would start from x = 0 and calculate the electric fields at different distances, such as x = 0.1 m, 0.2 m, 0.3 m, and so on, until you find a point where the magnitudes of the electric fields due to charges A and B are equal.