Question

a charge ( q = 13.2 , mu\text{c} ) is placed on a conducting spherical shell with inner radius ( r_1 = 9.00 , \text{cm} ) and outer radius ( r_2 = 10.0 , \text{cm} ). a point charge ( q = 9.84 , mu\text{c} ) is placed at the center of the cavity. the magnitude of the electric field at a point a distance ( r = 6.09 , \text{cm} ) from the center is (____) n/c.

Explanation:

Step1: Identify the relevant charge for Gauss's law

The point is at \( r = 6.09\space cm \), which is less than the inner radius \( R_1 = 9.00\space cm \) of the conducting shell. Inside the cavity (where \( r < R_1 \)), the electric field is determined by the point charge \( q \) at the center (the charge \( Q \) on the conducting shell does not contribute to the electric field inside the cavity of a conductor). So we use \( q = 9.84\space \mu C=9.84\times 10^{- 6}\space C \), and \( r = 6.09\space cm = 0.0609\space m \).

Step2: Apply Gauss's law for electric field

The formula for the electric field due to a point charge is \( E=\frac{kq}{r^{2}} \), where \( k = 8.988\times 10^{9}\space N\cdot m^{2}/C^{2} \).

Substitute the values:
\( E=\frac{8.988\times 10^{9}\times9.84\times 10^{-6}}{(0.0609)^{2}} \)

First, calculate the numerator: \( 8.988\times 10^{9}\times9.84\times 10^{-6}=8.988\times9.84\times 10^{3}\approx88.44\times 10^{3} = 8.844\times 10^{4} \)

Then, calculate the denominator: \( (0.0609)^{2}=0.0609\times0.0609\approx0.00371 \)

Now, divide the numerator by the denominator: \( E=\frac{8.844\times 10^{4}}{0.00371}\approx2.384\times 10^{7}\space N/C \)

Answer:

\( \approx2.38\times 10^{7}\space N/C \) (or more precise value depending on calculation steps)