An unknown charge sits on a conducting spherical shell of radius 10 cm. If the electric field 15 cm from the center of the sphere has the magnitude 3.0E3 N/C and is directed radially inward, what is the net charge on the sphere?
To find the net charge on the conducting spherical shell, we can use Gauss's law.
Gauss's law states that the electric flux through a closed surface is equal to the net charge enclosed by that surface divided by the permittivity of free space (ε₀).
In this case, we'll consider a Gaussian surface in the shape of a sphere centered at the center of the conducting shell and with a radius of 15 cm. Since the electric field is directed radially inward, the electric flux through this Gaussian surface will be negative.
We can calculate the electric flux (Φ) using the formula:
Φ = E * A,
where E is the electric field magnitude and A is the area of the Gaussian surface.
The area of a sphere can be calculated using the formula:
A = 4πr²,
where r is the radius of the Gaussian surface.
Substituting the values given:
E = 3.0E3 N/C,
r = 15 cm = 0.15 m,
we can calculate the electric flux:
Φ = E * A
= 3.0E3 N/C * 4π * (0.15 m)².
Using a calculator, we find:
Φ ≈ -2827.43 N·m²/C.
According to Gauss's law, the electric flux through a closed surface equals the net charge enclosed by that surface divided by ε₀:
Φ = Q / ε₀,
where Q is the net charge.
The permittivity of free space, ε₀, is approximately 8.85E-12 N·m²/C².
We can rearrange this equation to solve for the net charge Q:
Q = Φ * ε₀.
Substituting the calculated values:
Q ≈ -2827.43 N·m²/C * 8.85E-12 N·m²/C².
Using a calculator, we find:
Q ≈ -2.50E-8 C.
The negative sign indicates that the net charge on the conducting spherical shell is negative. Therefore, the net charge on the sphere is approximately -2.50E-8 Coulombs.
To find the net charge on the conducting spherical shell, we can use Gauss's Law. Gauss's Law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the net charge enclosed by that surface.
Step 1: Understand the problem
In this case, we are given the electric field magnitude (E) at a certain distance from the center of the sphere and the radial direction of the field. We want to find the net charge on the spherical shell.
Step 2: Recall Gauss's Law
Gauss's Law states that the electric flux (Φ) through a closed surface is equal to the net charge (Q) enclosed by that surface divided by the electric constant (ε₀):
Φ = Q / ε₀
Step 3: Relate electric flux to electric field
Using Gauss's Law, we can relate the electric flux through a closed surface to the electric field passing through that surface. For a spherical surface, the electric flux is given by:
Φ = E * A
where E is the electric field magnitude and A is the area of the spherical surface.
Step 4: Calculate the area of the closed surface
The spherical surface in this problem is defined by a radial distance of 15 cm from the center of the sphere. To calculate the surface area (A) of a sphere with radius r, we use the formula:
A = 4πr²
Step 5: Calculate the electric flux
Using the given electric field magnitude (E = 3.0E3 N/C) and the area of the closed surface (A = 4π(15E-2)²), we can calculate the electric flux (Φ) through the surface.
Step 6: Calculate the net charge
Once we have the electric flux, we can rearrange Gauss's Law and solve for the net charge (Q):
Q = Φ * ε₀
Step 7: Plug in the values and solve for the net charge
Now, we can substitute the calculated electric flux (Φ) and the value of the electric constant (ε₀ = 8.85E-12 F/m) into the equation to find the net charge (Q).
Following these steps, you can calculate the net charge on the conducting spherical shell based on the given information.