Two spherical shells have a common center. A -3.00 × 10-6 C charge is spread uniformly over the inner shell, which has a radius of 0.050 m. A +4.00 × 10-6 C charge is spread uniformly over the outer shell, which has a radius of 0.15 m. Find the magnitude and direction of the electric field at a distance (measured from the common center) of 0.10 m,
To find the magnitude and direction of the electric field at a distance of 0.10 m from the common center, we can use the principle of superposition.
Here are the steps to calculate the electric field:
Step 1: Calculate the electric field from each shell separately.
- The electric field due to a uniformly charged spherical shell is given by the equation:
E = (k * Q) / r^2
where E is the electric field, k is the electrostatic constant (9.0 x 10^9 Nm^2/C^2), Q is the charge on the shell, and r is the distance between the center of the shell and the point where you want to find the electric field.
Step 2: Calculate the electric field due to the inner shell.
- The inner shell has a negative charge of -3.00 x 10^-6 C, and its radius is 0.050 m.
- Using the formula from step 1, we can calculate the electric field from the inner shell at a distance of 0.10 m:
E_inner = (k * Q_inner) / r_inner^2
where Q_inner is the charge on the inner shell and r_inner is the distance from the center to the point where you want to find the electric field.
Step 3: Calculate the electric field due to the outer shell.
- The outer shell has a positive charge of +4.00 x 10^-6 C, and its radius is 0.15 m.
- Using the formula from step 1, we can calculate the electric field from the outer shell at a distance of 0.10 m:
E_outer = (k * Q_outer) / r_outer^2
where Q_outer is the charge on the outer shell and r_outer is the distance from the center to the point where you want to find the electric field.
Step 4: Add the electric fields from both shells to get the total electric field.
- Since the electric field is a vector quantity, we need to consider the direction as well.
- The direction of the electric field due to the inner shell is radially inward towards the center.
- The direction of the electric field due to the outer shell is radially outward away from the center.
- Since the charges are spread uniformly over each shell, the symmetry implies that the electric fields add up as scalar quantities.
Step 5: Calculate the net electric field at the given distance by adding the magnitudes of the electric fields and assigning the direction based on the sign of the sum.
- The net electric field will be the sum of the magnitudes of the electric fields due to the inner and outer shells.
- If the resultant sum is positive, the direction will be radially outward. If the resultant sum is negative, the direction will be radially inward.
Using these steps, we can calculate the magnitude and direction of the electric field at a distance of 0.10 m from the common center.