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.