A conductor in a shape of a spherical shell of radii 10 and 12cm in charged to 20 micro coloumb.
1) Find the electric field at the following radii: 2,11,15 cm.
2)Find the electric potential at each of these radii.
Repeat part 1 and a -5 microcoloumb point charge is being placed at the center of the sphere.
To find the electric field at a specific radius, we can use Gauss's Law, which states that the electric flux through any closed surface is equal to the charge enclosed divided by the permittivity of free space (ε₀). In this case, since the conductor is a spherical shell, the electric field inside the conductor is zero, and the charge is uniformly distributed on the outer surface of the shell.
1) To find the electric field at the radii of 2, 11, and 15 cm, we need to consider whether each radius is inside the shell (r < 10 cm), between the inner and outer radii (10 cm < r < 12 cm), or outside the shell (r > 12 cm).
a) For r = 2 cm: Since 2 cm < 10 cm, this radius is inside the shell. The electric field inside the conductor is zero, so the electric field at this radius is also zero.
b) For r = 11 cm: Since 10 cm < 11 cm < 12 cm, this radius is between the inner and outer radii of the shell. In this case, we can assume that the charge is concentrated at the center of the shell, forming a point charge. The electric field due to a point charge can be found using Coulomb's Law:
Electric Field (E) = (1 / (4πε₀)) * (Q / r²)
Substituting the values, we get:
Electric Field (E) = (1 / (4πε₀)) * (20μC / (0.11 m)²)
Note that we converted centimeters to meters, which gives us 0.11 m.
c) For r = 15 cm: Since 15 cm > 12 cm, this radius is outside the shell. The electric field outside a uniformly charged shell is the same as that of a point charge located at the center of the shell. Therefore, we can use the same formula as in part b to find the electric field at this radius.
2) To find the electric potential at each of these radii, we use the formula:
Electric Potential (V) = (1 / (4πε₀)) * (Q / r)
Where Q is the total charge on the shell.
a) For r = 2 cm: Since this radius is inside the shell, the electric potential is zero.
b) For r = 11 cm: Using the same formula as before, we get:
Electric Potential (V) = (1 / (4πε₀)) * (20μC / (0.11 m))
c) For r = 15 cm: The formula remains the same, giving us:
Electric Potential (V) = (1 / (4πε₀)) * (20μC / (0.15 m))
Repeat part 1 and consider that a -5 μC point charge is placed at the center of the sphere:
1) Since there is now a point charge at the center of the sphere, the electric field at any radius will depend on the combined effects of the charged shell and the point charge.
a) For r = 2 cm: The electric field at this radius is no longer zero, as the point charge contributes to the overall electric field. To find the total electric field, we need to sum the electric field due to the charged shell and the point charge. The electric field due to the point charge can be calculated using Coulomb's Law, and the electric field due to the shell can be found using Gauss's Law as explained in part 1.
b) For r = 11 cm: The same procedure as in part a is followed to find the total electric field at this radius.
c) For r = 15 cm: The same procedure as in part a is followed to find the total electric field at this radius.
To find the electric potential at each of these radii, we can use the same formula as in part 2:
a) For r = 2 cm: The electric potential at this radius will also depend on the combined effects of the charged shell and the point charge. We need to sum the electric potential due to the charged shell and the point charge.
b) For r = 11 cm: The same procedure as in part a is followed to find the total electric potential at this radius.
c) For r = 15 cm: The same procedure as in part a is followed to find the total electric potential at this radius.