A solid sphere with radius a = 2 cm has a total charge Q = 10 nC which is uniformly distributed throughout its volume. Using Gauss’s Law, determine the magnitude of the electric field at a point 1 cm from the center of the sphere.Multiple Choice1.12 × 105 N/C1.12 × 105 N/C2.25 × 105 N/C2.25 × 105 N/C0 N/C3.61 × 105 N/C
Question
A solid sphere with radius a = 2 cm has a total charge Q = 10 nC which is uniformly distributed throughout its volume. Using Gauss’s Law, determine the magnitude of the electric field at a point 1 cm from the center of the sphere.Multiple Choice1.12 × 105 N/C1.12 × 105 N/C2.25 × 105 N/C2.25 × 105 N/C0 N/C3.61 × 105 N/C
Solution
To solve this problem, we will use Gauss's Law, which states that the total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity of free space.
Step 1: Identify the given values. The radius of the sphere, a = 2 cm = 0.02 m The charge of the sphere, Q = 10 nC = 10 x 10^-9 C The distance from the center of the sphere, r = 1 cm = 0.01 m The permittivity of free space, ε0 = 8.85 x 10^-12 C^2/N*m^2
Step 2: Determine the volume of the sphere. The volume of a sphere is given by V = 4/3πr^3. Since we are looking at a point inside the sphere, we use the smaller radius r = 0.01 m. So, V = 4/3π(0.01 m)^3 = 4.19 x 10^-6 m^3.
Step 3: Determine the charge enclosed within the smaller sphere. The charge density ρ is given by the total charge Q divided by the total volume of the sphere. So, ρ = Q/V = (10 x 10^-9 C) / (4/3π(0.02 m)^3) = 0.596 x 10^-3 C/m^3. The charge enclosed q within the smaller sphere is then given by q = ρV = (0.596 x 10^-3 C/m^3) * (4/3π(0.01 m)^3) = 0.25 x 10^-9 C.
Step 4: Apply Gauss's Law. The electric field E at a distance r from the center of a sphere with enclosed charge q is given by E = q / (4πε0r^2). So, E = (0.25 x 10^-9 C) / (4π(8.85 x 10^-12 C^2/N*m^2)(0.01 m)^2) = 2.24 x 10^5 N/C.
So, the magnitude of the electric field at a point 1 cm from the center of the sphere is approximately 2.24 x 10^5 N/C. This is not exactly one of the given options, but it is closest to the second option, 2.25 x 10^5 N/C.
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