A point charge (m = 1.0 g) at the end of an insulating string of length L = 51 cm (Fig. 16-66) is observed to be in equilibrium in a uniform horizontal electric field of E = 9200 N/C, when the pendulum's position is as shown in Fig. 16-66, with the charge d = 2.0 cm above the lowest (vertical) position. If the field points to the right in Fig. 16-66, determine the magnitude and sign of the point charge.
To determine the magnitude and sign of the point charge, we can use the concept of equilibrium in an electric field.
Given information:
Mass of the point charge, m = 1.0 g = 0.001 kg
Length of the insulating string, L = 51 cm = 0.51 m
Vertical displacement of the charge, d = 2.0 cm = 0.02 m
Electric field, E = 9200 N/C (points to the right in the figure)
In equilibrium, the gravitational force acting downwards is balanced by the electric force acting upwards. The gravitational force can be calculated using the formula:
F_g = mg
Here, g is the acceleration due to gravity. In this problem, since the charge is in equilibrium, the gravitational force F_g is balanced by the electric force F_e.
The electric force can be calculated using the formula:
F_e = qE
Here, q is the charge and E is the electric field.
Since the charge is in equilibrium, the horizontal components of the tension in the string and the electric force should be equal.
Let's calculate the tension in the string using the formula:
T = mgcosθ
Given the vertical displacement, d, we can calculate the angle, θ, using the formula:
sinθ = d/L
Once we have the tension, T, we can equate it to the electric force, Fe, and solve for the charge, q.
T = Fe
mgcosθ = qE
Substituting the given values into the equation, we can solve for q.
q = (mgcosθ) / E
Calculate the value of g, θ, cosθ, and substitute these values along with the given values to find q. The magnitude of the charge will be positive or negative depending on the direction of the electric field.