Question

A charged sphere is suspended from a string in a uniform electric field directed horizontally. There is an electric force on the sphere to the right and a gravitational force pointing downward. As a result, the sphere hangs at an angle theta from the vertical. Combinations of sphere mass and electric charge are listed in the chart for four cases, all in the same uniform electric field. Rank the angle theta that the string forms with the vertical for these different spheres. Find a mathematical expression for the angle theta. Then strategize on how to get the ranking with only the barest minimum of simple calculations, not using a calculator.

Answer 1

The net force on charged sphere along x - direction on its zero. T sin theta - q E = 0 T sin theta = q E And the net force on charged sphere along y - direction is zero. T cos theta - mg = 0 T cos theta = mg The angle theta is theta = tan^- 1 (q E/mg) In the case (B), q/m = 8 Times 10^- 9 C/3 Times 10^- 3 kg = 2.67 Times 10^- 6 C/kg In the case (B), q/m = 4 Times 10^- 9 C/6 Times 10^- 3 kg = 0.067 Times 10^- 6 C/kg In the case (C), q/m = 2 Times 10^- 9 C/9 Times 10^- 3 kg = 0.22 Times 10^- 6 C/kg In the case (D), q/m = 8 Times 10^- 9 C/6 Times 10^- 3 kg = 1.33 Times 10^- 6 C/kg As the value of q/m is large, the angle theta is also large. So the rank of angle in ascending order is theta_C, theta_B, theta_D, and theta_A.