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. This principle states that the total electric field at a point is the vector sum of the electric fields due to each individual charge.

Let's break down the problem into smaller steps:

Step 1: Calculate the electric field due to the inner shell charge:
Using the formula for the electric field of a uniformly charged sphere, we have:
E_inner = (k * Q_inner) / r^2

Where:
E_inner is the electric field due to the inner shell charge
k is the Coulomb's constant (k = 9.0 x 10^9 Nm^2/C^2)
Q_inner is the charge on the inner shell (-3.00 x 10^-6 C)
r is the distance from the common center to the point of interest (0.10 m)

Plugging in the values, we get:
E_inner = (9.0 x 10^9 Nm^2/C^2 * (-3.00 x 10^-6 C)) / (0.10 m)^2

Step 2: Calculate the electric field due to the outer shell charge:
Similarly, using the same formula, we have:
E_outer = (k * Q_outer) / r^2

Where:
E_outer is the electric field due to the outer shell charge
Q_outer is the charge on the outer shell (+4.00 x 10^-6 C)

Plugging in the values, we get:
E_outer = (9.0 x 10^9 Nm^2/C^2 * (4.00 x 10^-6 C)) / (0.10 m)^2

Step 3: Calculate the total electric field:
The total electric field at the point is the vector sum of the electric fields due to the inner and outer shells:
E_total = E_inner + E_outer

Step 4: Determine the magnitude and direction:
To find the magnitude of the electric field, take the absolute value of E_total:
|E_total| = |E_inner + E_outer|

To find the direction of the electric field, we need to consider the signs of the charges. If the charges are opposite, then the direction of the electric field will be away from the positive charge and towards the negative charge. If the charges are the same, the direction will be from positive to positive or from negative to negative.

Now, you can plug in the values and calculate the electric field at a distance of 0.10 m from the common center.