A positive charge Q is on the y-axis at a distance a from the origin, and another positive charge q is on the x-axis at a distance b from the origin.

A) For what values(s) of b is the x-component of the force on q a minimum?
B) For what values(s) of b is the x-component of the force on q a maximum?

To determine the x-component of the force on q, we need to calculate the electric field at q due to the charge Q. The electric field at q is given by:

E = k * Q / r^2

where k is the electrostatic constant, Q is the charge on the y-axis, and r is the distance between the charges.

Let's calculate the x-component of the force on q using this formula:

F_x = q * E * cos(θ)

where q is the charge on the x-axis, E is the electric field, and θ is the angle between the electric field and the x-axis.

A) To find the value(s) of b for which the x-component of the force on q is a minimum, we need to find the angle θ at which cos(θ) is a minimum. Since cos(θ) is a minimum when θ = 180 degrees (π radians), we need to find the value(s) of b for which the two charges are aligned along the y-axis. In other words, when q is directly above Q on the y-axis.

Therefore, the value(s) of b for which the x-component of the force on q is a minimum is when b = 0.

B) To find the value(s) of b for which the x-component of the force on q is a maximum, we need to find the angle θ at which cos(θ) is a maximum. Since cos(θ) is a maximum when θ = 0 degrees (0 radians), we need to find the value(s) of b for which the two charges are aligned along the x-axis. In other words, when q is directly to the right of Q on the x-axis.

Therefore, the value(s) of b for which the x-component of the force on q is a maximum is when b = a.

when b is zero, there is no x component

Now, the x component of force is

Fx=kqq/(a+b) * cosTheta

where cosTheta= b/(sqrt(a+b))

so, take the deriviative of Fx with respect to b, and solve.