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 = 1.0 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.
up 1 cm on 51 cm string
cos A = 50/51
sin A = side force F/ Tension T
tan A = F/(mg)
so
F/mg = tan 11.36 = .201
F = .001 (9.81)(.201) = .00197 Newtons
F = Q E
.00197 = Q * 9200
Q = 2.14 * 10^-7 Coulomb
If motion in direction of E field, charge positive
To determine the magnitude and sign of the point charge, we can use the concept of equilibrium in the presence of electric forces.
In this scenario, the point charge is in equilibrium, which means that the electric force acting on the charge is balanced by the gravitational force.
To find the magnitude of the point charge, we can calculate the gravitational force acting on the charge. The gravitational force is given by the formula:
F_grav = m * g
where m is the mass of the charge and g is the acceleration due to gravity.
In this case, we are given that the mass (m) of the charge is 1.0 g. To convert this to kg, we divide by 1000:
m = 1.0 g / 1000 = 0.001 kg
The acceleration due to gravity (g) is approximately 9.8 m/s².
So, the gravitational force acting on the charge is:
F_grav = 0.001 kg * 9.8 m/s² = 0.0098 N
Next, we need to find the electric force acting on the charge. The electric force is given by the formula:
F_electric = q * E
where q is the charge and E is the electric field.
In this scenario, the electric field (E) is given as 9200 N/C. We need to determine the charge (q).
To find the charge, we can consider the fact that the charge is in equilibrium when it is displaced d = 1.0 cm above the lowest position. At this point, the gravitational force and the electric force are equal in magnitude and opposite in direction:
|F_grav| = |F_electric|
Therefore:
0.0098 N = q * 9200 N/C
Now we can solve for q:
q = 0.0098 N / 9200 N/C
q = 1.06 x 10^-6 C
Finally, we determine the magnitude and sign of the point charge.
The magnitude of the charge is given by:
|q| = 1.06 x 10^-6 C
Since the field points to the right in the figure, and the charge is in equilibrium when displaced above the lowest position, we can conclude that the charge is negative.
Therefore, the magnitude of the charge is 1.06 x 10^-6 C, and the charge is negative.
To determine the magnitude and sign of the point charge, we can use the equilibrium condition for charges under the influence of an electric field. The gravitational force acting on the point charge is balanced by the electric force due to the electric field.
Step 1: Find the weight of the point charge
The weight of the point charge can be calculated using the formula:
Weight = mass * acceleration due to gravity
Given:
mass (m) = 1.0 g = 0.001 kg
acceleration due to gravity (g) = 9.8 m/s^2
Weight = 0.001 kg * 9.8 m/s^2
Weight = 0.0098 N
Step 2: Find the electric force on the point charge
The electric force on the charge can be calculated using the formula:
Electric force = charge * electric field
Given:
electric field (E) = 9200 N/C
distance from the lowest position (d) = 1.0 cm = 0.01 m
Electric force = charge * 9200 N/C
Step 3: Set up the equilibrium equation
For equilibrium, the electric force must balance the weight of the charge. Therefore, we have:
Electric force = Weight
Charge * 9200 N/C = 0.0098 N
Step 4: Calculate the charge
Solving for the charge:
Charge = 0.0098 N / 9200 N/C
Charge = 1.0652 x 10^-6 C
The magnitude of the charge is approximately 1.0652 x 10^-6 C.
Step 5: Determine the sign of the charge
Since the given electric field points to the right in the figure, and the point charge is displaced upward to reach equilibrium, it must be a positive charge.
Therefore, the magnitude of the point charge is approximately 1.0652 x 10^-6 C, and it is a positive charge.