A hollow metal sphere has inner and outer radii of 20.0 cm and 30.0 cm, respectively. As shown in the figure, a solid metal sphere of radius 10.0 cm is located at the center of the hollow sphere. The electric field at a point P, a distance of 15.0 cm from the center, is found to be E1 = 1.42 104 N/C, directed radially inward. At point Q, a distance of 35.0 cm from the center, the electric field is found to be E2 = 1.32 104 N/C, directed radially outward. Determine the total charge on the surface of the inner sphere, the inner surface of the hollow sphere, and the outer surface of the hollow sphere.
To determine the total charge on the surface of the inner sphere, the inner surface of the hollow sphere, and the outer surface of the hollow sphere, we can make use of Gauss's law and the properties of electric fields.
Gauss's law states that the electric flux through a closed surface is directly proportional to the charge enclosed by the surface. In this case, we can imagine drawing a Gaussian surface around each sphere, such that the electric field lines pass through the surface.
Let's start by calculating the total charge on the surface of the inner sphere. Since the electric field at point P, which is inside the inner sphere, is directed radially inward, it means that there is a negative charge enclosed by the Gaussian surface, creating an electric field directed inward. Therefore, the charge on the surface of the inner sphere is negative.
Using Gauss's law, we can write the equation:
Φ = q_enclosed / ε₀
Where Φ is the electric flux through the Gaussian surface, q_enclosed is the charge enclosed by the surface, and ε₀ is the electric constant.
Since the electric field at point P is E1 = 1.42 * 10^4 N/C, and the radius of the inner sphere is 10.0 cm, we can use the formula for the electric field of a point charge to find the charge enclosed by the Gaussian surface.
E1 = k * (q_enclosed / r^2)
Rearranging the equation to solve for the charge:
q_enclosed = E1 * r^2 / k
Plugging in the values:
q_enclosed = (1.42 * 10^4 N/C) * (0.1 m)^2 / (8.99 * 10^9 Nm^2/C^2)
Calculating it, we find:
q_enclosed = -1.586 C
Therefore, the total charge on the surface of the inner sphere is approximately -1.586 Coulombs.
Now, let's move on to the inner surface of the hollow sphere. Since the electric field at point Q, which is inside the hollow sphere, is directed radially outward, it means that there is a positive charge enclosed by the Gaussian surface, creating an electric field directed outward. Therefore, the charge on the inner surface of the hollow sphere is positive.
Using the same process, we can use Gauss's law and the electric field at point Q to find the charge enclosed by the Gaussian surface.
E2 = k * (q_enclosed / r^2)
Rearranging the equation to solve for the charge:
q_enclosed = E2 * r^2 / k
Plugging in the values:
q_enclosed = (1.32 * 10^4 N/C) * (0.3 m)^2 / (8.99 * 10^9 Nm^2/C^2)
Calculating it, we find:
q_enclosed = 1.32 C
Therefore, the total charge on the inner surface of the hollow sphere is approximately 1.32 Coulombs.
Lastly, let's determine the charge on the outer surface of the hollow sphere. Since the electric field inside a metal is zero, the net charge inside the hollow sphere must be zero. Therefore, the total positive charge on the outer surface of the hollow sphere must balance out the total negative charge on the inner surface.
The charge on the outer surface is given by the sum of the charge on the inner surface and the charge on the surface of the inner sphere:
q_outer = q_inner_surface + q_inner_sphere_surface
Plugging in the values:
q_outer = 1.32 C + (-1.586 C)
Calculating it, we find:
q_outer = -0.266 C
Therefore, the total charge on the outer surface of the hollow sphere is approximately -0.266 Coulombs.
In summary:
- The total charge on the surface of the inner sphere is approximately -1.586 Coulombs.
- The total charge on the inner surface of the hollow sphere is approximately 1.32 Coulombs.
- The total charge on the outer surface of the hollow sphere is approximately -0.266 Coulombs.