Three small spheres are placed at fixed points along the x-axis, whose positive direction points towards the right.

Sphere A is at x = 23.0 cm, with a charge of -10.00 microCoulomb
Sphere B is at x = 55.0 cm, with a charge of 2.00 microCoulomb
Sphere C is at x = 57.0 cm, with a charge of -9.00 microCoulomb

(A)If Sphere B is removed. Give the x-coordinate of the point on the x-axis where the field due to spheres A and C is zero? (answer 40.4 cm)

The point where the fields due to A and C cancel has nothing to do with the size of charge B. It also does not depend upon where charge B started out. The point will be between A and C. Let its X-coordinate (in cm) be x. The Coulomb constant k will cancel out, and you can use cm for x and uC for charge. You end up solving this equation:

10/(x-23)^2 = 9/(57-x)^2

(57-x) = sqrt(9/10)*(x-23)
= 0.9487*(x-23)
57 + 21.82 = 1.9487 x
x = ___ cm

To find the x-coordinate of the point on the x-axis where the electric field due to spheres A and C is zero, we can use the principle of superposition.

The electric field at a point is the vector sum of the electric fields due to each individual charge. When the electric field is zero at a certain point, it means that the vector sum of the electric fields due to Sphere A and Sphere C cancels out.

Let's assume that the x-coordinate of the point where the electric field is zero is x = d. The electric field due to Sphere A at the point x = d can be calculated using Coulomb's law:

E_A = k * |q_A| / r_A^2

Where:
- E_A is the electric field due to Sphere A
- k is the electrostatic constant (k = 8.99 x 10^9 Nm^2/C^2)
- |q_A| is the magnitude of the charge of Sphere A
- r_A is the distance between Sphere A and the point x = d

Similarly, the electric field due to Sphere C at the point x = d can be calculated using the same formula.

We want to find the value of d, where the electric field due to Sphere A cancels out the electric field due to Sphere C. In other words:

E_A - E_C = 0

Substituting the expressions for E_A and E_C, we get:

k * |q_A| / r_A^2 - k * |q_C| / r_C^2 = 0

Now, let's substitute the given values:

k * 10.00 μC / (d - 23.0 cm)^2 - k * 9.00 μC / (d - 57.0 cm)^2 = 0

Simplifying this equation:

10.00 / (d - 23.0)^2 = 9.00 / (d - 57.0)^2

Now, we can solve this equation to find the value of d. Since it is a quadratic equation, we can simplify it further and use numerical methods (such as using a graphing calculator or solving numerically) to find the value of d.

After solving the equation, we find that the x-coordinate of the point where the electric field due to Sphere A and Sphere C is zero is approximately 40.4 cm.