A solid metal sphere with radius 0.430{\rm m} carries a net charge of 0.210{\rm nC} .
Find the magnitude of the electric field at a point 0.112{\rm m} outside the surface of the sphere.
Find the magnitude of the electric field at a point inside the sphere, 0.112{\rm m} below the surface.
To find the magnitude of the electric field at a point outside the surface of the sphere, we can use Gauss's Law. Gauss's Law relates the electric field with the charge enclosed by a Gaussian surface. Here are the steps to find the electric field:
Step 1: Calculate the charge enclosed by a Gaussian surface.
The entire charge of the sphere is given as 0.210 nC.
Step 2: Determine the area of the Gaussian surface.
The Gaussian surface in this case is a concentric spherical surface outside the sphere. The area of the Gaussian surface is calculated using the formula for the surface area of a sphere:
Area = 4πr^2
where r is the distance between the point and the center of the sphere.
Step 3: Apply Gauss's Law.
Gauss's Law states that the electric flux through a closed surface is proportional to the charge enclosed by the surface. In equation form, it is given by:
Φ = Q_enclosed / ε_0
where Φ is the electric flux, Q_enclosed is the charge enclosed by the surface, and ε_0 is the permittivity of free space.
Since the electric field is constant on a Gaussian surface, we can equate the electric flux to E times the area of the Gaussian surface:
Φ = E * Area
Step 4: Solve for the electric field.
Rearrange the equation to solve for E:
E = Φ / Area
Now let's plug in the numbers:
Charge enclosed: 0.210 nC
Radius of the Gaussian surface: 0.430 m + 0.112 m (distance outside the surface)
Area of the Gaussian surface: 4π(0.430 + 0.112)^2
permittivity of free space (ε_0): 8.854 x 10^-12 C^2/(N.m^2)
Calculating the electric field at a point 0.112 m outside the surface:
E = (0.210 nC) / [4π(0.430 + 0.112)^2] / (8.854 x 10^-12)
E = 9243.46 N/C
To find the electric field at a point inside the sphere, 0.112 m below the surface, we can use a similar approach. However, since the point is inside the sphere, the electric field inside the sphere is not influenced by the additional charge.
The electric field at a point inside a charged solid sphere is given by:
E = k * (Q / r^2)
where k is the electrostatic constant (9 x 10^9 N.m^2/C^2), Q is the total charge on the sphere, and r is the distance from the center of the sphere.
To calculate the electric field at a point 0.112 m below the surface, we can use the following values:
Charge on the sphere: 0.210 nC
Radius of the sphere: 0.430 m
Distance from the center of the sphere: 0.430 m - 0.112 m
Plugging in these values:
E = (9 x 10^9) * (0.210 x 10^-9) / (0.318 m)^2
E = 271,067.29 N/C
Therefore, the magnitude of the electric field at a point 0.112 m outside the surface of the sphere is 9243.46 N/C, and the magnitude of the electric field at a point inside the sphere, 0.112 m below the surface, is 271,067.29 N/C.