A sphere with a mass m and a net charge Q is tied to a string and placed in a uniform electric field E. Once the sphere has come to equilibrium, the string makes an angle theta. Find the charge Q in terms of E, m, and theta.
I am not sure how to combine the force of tension and the force of gravity, and how that relates to charge.
Thank you.
To find the charge Q in terms of E, m, and theta, we need to analyze the forces acting on the sphere in equilibrium.
1. The force of gravity (Fg) acts downward and can be calculated using Newton's second law: Fg = mg, where g is the acceleration due to gravity.
2. The force of tension in the string (Ft) acts upward at an angle theta to balance the force of gravity. This force is given by Ft = mgcos(theta).
3. The electric force (Fe) due to the net charge Q in the uniform electric field E acts in the direction of the field. The magnitude of this force is given by Fe = QE.
In equilibrium, the forces in the vertical direction must balance. Therefore, we have:
Fg - Ft = 0
Substituting the expressions for Fg and Ft:
mg - mgcos(theta) = 0
mg(1 - cos(theta)) = 0
Dividing both sides by mg:
1 - cos(theta) = 0
cos(theta) = 1
Since the sphere is in equilibrium, the tension in the string exactly balances the force of gravity, and there is no net force acting in the vertical direction.
Now, let's consider the electric force. In equilibrium, the electric force will also balance the force of gravity. Therefore, we have:
Fe - Fg = 0
Substituting the expressions for Fe and Fg:
QE - mg = 0
Solving for Q:
Q = mg / E
Thus, the charge Q in terms of E, m, and theta is Q = mg / E.
To find the charge Q in terms of E, m, and theta, we need to consider the equilibrium condition of the sphere in the electric field.
Let's analyze the forces acting on the sphere in equilibrium:
1. Electric force (Fe): The electric field E exerts a force on the charged sphere, which is given by Fe = Q * E. This force acts in the direction of the electric field.
2. Tension force (Ft): The tension force in the string opposes the electric force and keeps the sphere in equilibrium. The tension force acts along the string.
3. Gravitational force (Fg): The mass of the sphere m experiences a gravitational force given by Fg = m * g, where g is the acceleration due to gravity. This force acts vertically downwards.
In equilibrium, these three forces balance each other, so we have:
Fe = Ft + Fg
Now, let's resolve the forces into their components:
1. The electric force Fe has only one component in the direction of the string, which is Fe*sin(theta).
2. The tension force Ft also has a component perpendicular to the string, which is Ft*cos(theta).
3. The gravitational force Fg has a component along the string, which is Fg*sin(theta).
Now substituting the components in the equilibrium equation, we have:
Q * E * sin(theta) = Ft * cos(theta) + m * g * sin(theta)
Simplifying, we can rearrange the equation to solve for the charge Q:
Q = (Ft * cos(theta) + m * g * sin(theta)) / (E * sin(theta))
Thus, the charge Q can be expressed in terms of E, m, and theta using the above equation.