A point charge \left( {m = 1.0\,{\rm{g}}} \right) at the end of an insulating cord of length 55 {\rm cm} is observed to be in equilibrium in a uniform horizontal electric field of 15,000\,{\rm{N/C}}, when the pendulum's position, with the charge 12 {\rm cm} above the lowest (vertical) position.
If the field points to the right in the figure, determine the magnitude of the point charge.
To analyze this situation, we can use the concept of equilibrium for a charged particle in an electric field. The gravitational force acting on the charge is balanced by the electric force due to the electric field.
To find the magnitude of the charge, we can use the equation for the force on a charge in an electric field:
F_electric = q * E
Where:
F_electric is the electric force acting on the charge
q is the charge on the particle
E is the electric field strength
Given that the electric field strength is 15,000 N/C and the charge is in equilibrium, we can equate the gravitational force and the electric force:
m * g = q * E
Where:
m is the mass of the charge (1.0 g = 0.001 kg)
g is the acceleration due to gravity (9.8 m/s^2)
Now, let's solve for the charge (q):
q = (m * g) / E
q = (0.001 kg * 9.8 m/s^2) / 15000 N/C
Calculating this, we find q ≈ 6.53 × 10^(-8) C.
To find the angle between the cord and the vertical, we can use the concept of equilibrium for a pendulum at a specific position. At the position where the charge is 12 cm above the lowest point, the tension in the cord (T) is equal to the component of the gravitational force (mg) perpendicular to the cord:
T = mg * cos(θ)
Where:
T is the tension in the cord
m is the mass of the charge (0.001 kg)
g is the acceleration due to gravity (9.8 m/s^2)
θ is the angle between the cord and the vertical
Given that the length of the cord is 55 cm, we can use trigonometry to find the value of cos(θ):
cos(θ) = 12 cm / 55 cm
cos(θ) ≈ 0.2182
Now, let's solve for the tension in the cord (T):
T = (0.001 kg * 9.8 m/s^2) * 0.2182
Calculating this, we find T ≈ 0.0021 N.
Therefore, the magnitude of the charge is approximately 6.53 × 10^(-8) C, and the tension in the cord is approximately 0.0021 N.