Question

A small sphere with mass 1.5g hangs by a thread between two parallel vertical plates 5cm apart. The plates are insulating and have uniform surface charges densities Q and -Q. The charge on the sphere is q=8.9*10-16C. What potential difference between the plates will cause the thread to assume an angle of 30o with the vertical?

Answer 1

The tralational equilibrium condition allows finding that the electric potential is   V = 4.8 10¹¹ V

Given parameter

• The mass m = 1.5 g = 1.15 10-3 kg

• The charge on the sphere q = 8.9 10-16 C

• Plate spacing d = 5 cm = 5.00 10-2 m

To find

• The potential difference

Newton's second law states that the force is proportional to the mass and the acceleration of the bodies, in the special case that the acceleration is zero, it establishes the condition for the equilibrium of the bodies

          ∑ F = 0

Where the bold indicate vector and F is the force

To use this equation we must fix a reference system with respect to which to carry out the decomposition and measurements of the forces; let's fix a system with the horizontal x axis and the vertical y axis, in the attachment I could see a free body diagram.

x- axis

     Fe - Tₓ = 0

     Fe = Tₓ

y-axis

     $T_y$ - W = 0

     W = $T_y$

     mg = $T_y$

The electric force is

      Fe = q E = q V / d

let's use trigonometry to decompose the stress

     cos 30 =  $T_y$ / T

     sin 30 = Tₓ / T

      $T_y$ = T cos 30

      Tₓ = T sin 30

We substitute

        q V / d = T sin 30

        mg = T cos 30

It's solve the system of equations

         $\frac{q \ V}{d \ m g}$ = tan 30

         V = $\frac{d \ mg }{ q}\ tan \ 30$

It's calculate

         V = $\frac{5.00 \ 10^{-2} 1.5 \ 10^{-3} \ 9.8}{ 8.9 10^{-16} } \ tan \ 30$

         V = 4.768 10¹¹ V

In conclusion, using the equilibrium condition, we could find that the electric potential is V = 4.8 10¹¹ V

Learn more about equilibrium condition here:

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