A box of mass m which carries a charge Q slides down a frictionless inclined plane that makes an angle, theta, with the horizontal. An electric field E points from right to left. How large must E be for the acceleration down the incline to be zero?
To determine the electric field E required for the acceleration down the incline to be zero, we need to consider the forces acting on the box.
1. Gravity force (Fg): The force due to gravity acts vertically downward and can be calculated using the formula Fg = m * g, where m is the mass of the box and g is the acceleration due to gravity (9.8 m/s^2).
2. Electric force (Fe): The electric force due to the electric field E acts horizontally and can be calculated using the formula Fe = Q * E. Here, Q is the charge on the box.
3. Normal force (Fn): The normal force acts perpendicular to the inclined plane and cancels the component of the gravity force acting in that direction.
4. Friction force (Ff): Since the inclined plane is frictionless, there is no friction force acting on the box.
For the acceleration down the incline to be zero, the sum of the forces acting in that direction must be zero. In this case, the forces acting down the incline are the component of the gravity force and the electric force.
The component of the gravity force down the incline can be calculated as Fg_parallel = m * g * sin(theta), where theta is the angle of the inclined plane.
Setting up the equation for the sum of forces down the incline:
Fg_parallel + Fe = 0
m * g * sin(theta) + Q * E = 0
Solving for E:
E = - (m * g * sin(theta)) / Q
Therefore, the magnitude of the electric field E required for the acceleration down the incline to be zero is given by E = (m * g * sin(theta)) / Q.