A square plate of copper with 50 cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicular to the plate. Find (a) thr charge density at each face of the plate and (b) the total charge on each face.
a) Use Gauss' Law for the surface charge density in coulombs/m^2.
The front and back surfaces will have equal and opposite charge densities, since the net charge is zero.
b) Multiply the answer in a) by the plate area, 0.25 m^2.
To find the charge density at each face of the plate, we need to first recall the equation for charge density:
Charge density = Total charge / Area
Given that the plate is square and has 50 cm sides, we can find its area using the formula for the area of a square:
Area = (side length)^2 = (50 cm)^2
Now, let's find the charge density at each face:
(a) Charge density at each face of the plate:
To find the charge density at each face, we need to know the total charge on each face, which we will calculate in part (b). The charge density on each face will be the total charge on that face divided by the area of the face.
(b) Total charge on each face of the plate:
The electric field applies a force on the charges in the plate, but since the plate has no net charge, the force on the positive charges exactly cancels the force on the negative charges. Hence, the plate remains in equilibrium.
The electric field due to the plate will induce a charge separation such that one side of the plate becomes positively charged, while the other side becomes negatively charged.
The total charge on each face of the plate will be the product of the charge density and the area of the face.
To find the charge density on each face, we need to know the electric field strength at that face. Since the electric field is uniform and directed perpendicular to the plate, the electric field strength is the same at both faces.
Given that the electric field strength is 80.0 kN/C, we can calculate the total charge on each face as follows:
Total charge = Electric field strength * Area
Total charge on each face = 80.0 kN/C * Area
Now, let's calculate the values:
Area = (50 cm)^2 = 2500 cm^2
Total charge on each face = 80.0 kN/C * 2500 cm^2
Note: If we are required to provide the answer in another unit such as coulombs (C), we need to convert the charge from coulombs to the desired unit, e.g., microcoulombs (μC) or nanocoulombs (nC), by using appropriate conversion factors.