Question

23.3 a charged isolated conductor a charge q = 11.0 μc is placed on a conducting spherical shell with inner radius r₁ = 9.00 cm and outer radius r₂ = 10.0 cm. a point charge q = 7.26 μc is placed at the center of the cavity. the magnitude of the electric field at a point a distance r = 11.7 cm from the center is: n/c.

Explanation:

Step1: Identify the relevant charge for electric field

The point is at \( r = 11.7\space cm \), which is outside the conducting spherical shell (outer radius \( R_2 = 10.0\space cm \)). For a conducting shell with a charge \( Q \) on it and a point charge \( q \) inside the cavity, the electric field outside the shell is determined by the total charge enclosed, which is \( Q + q \) (by Gauss's law, as the shell's charge and the inner charge will contribute to the field outside). First, convert charges to coulombs and distance to meters.

\( Q = 11.0\space \mu C = 11.0\times 10^{- 6}\space C \)

\( q = 7.26\space \mu C = 7.26\times 10^{-6}\space C \)

\( r = 11.7\space cm = 0.117\space m \)

Step2: Apply Gauss's Law for electric field

The formula for electric field due to a point charge (or a charge distribution acting as a point charge outside) is \( E=\frac{k(Q + q)}{r^{2}} \), where \( k = 8.988\times 10^{9}\space Nm^{2}/C^{2} \)

First, calculate the total charge \( Q_{total}=Q + q=(11.0 + 7.26)\times 10^{-6}\space C=18.26\times 10^{-6}\space C \)

Then, substitute into the electric field formula:

\( E=\frac{8.988\times 10^{9}\times18.26\times 10^{-6}}{(0.117)^{2}} \)

Calculate the numerator: \( 8.988\times 10^{9}\times18.26\times 10^{-6}=8.988\times18.26\times 10^{3}\approx164.11\times 10^{3} \)

Calculate the denominator: \( (0.117)^{2}=0.013689 \)

Then \( E=\frac{164.11\times 10^{3}}{0.013689}\approx12.0\times 10^{6}\space N/C \) (more precisely, let's do the calculation:

\( 8.988\times18.26 = 8.988\times(18 + 0.26)=8.988\times18+8.988\times0.26 = 161.784+2.33688 = 164.12088 \)

\( 164.12088\times 10^{3}=164120.88 \)

\( 164120.88\div0.013689\approx12000000\space N/C = 1.20\times 10^{7}\space N/C \) (Wait, let's check the calculation again. Wait, \( r = 0.117\space m \), \( r^{2}=0.013689 \)

\( 164120.88\div0.013689 = 164120.88\div0.013689\approx12000000 \)? Wait, 0.013689\times12000000 = 164268, which is close to 164120.88. So approximately \( 1.20\times 10^{7}\space N/C \) or more accurately, let's compute:

\( E=\frac{8.988\times 10^{9}\times(11.0 + 7.26)\times 10^{-6}}{(0.117)^{2}}=\frac{8.988\times18.26\times 10^{3}}{0.013689} \)

\( 8.988\times18.26 = 164.12088 \)

\( 164.12088\times 10^{3}=164120.88 \)

\( 164120.88\div0.013689 = 164120.88\div0.013689\approx12000000 \) (since 0.013689*12000000 = 164268, which is very close to 164120.88, the difference is due to rounding in intermediate steps). So the electric field is approximately \( 1.20\times 10^{7}\space N/C \) or \( 12.0\times 10^{6}\space N/C \)

Wait, maybe I made a mistake in the total charge. Wait, the conducting shell has charge \( Q \), and the point charge \( q \) is inside the cavity. For a conducting shell, the charge on the inner surface of the shell will be \( - q \), and the charge on the outer surface will be \( Q + q \). So when we are outside the shell ( \( r > R_2 \) ), the electric field is due to the charge on the outer surface, which is \( Q + q \). So the total charge enclosed by a Gaussian sphere of radius \( r > R_2 \) is \( Q + q \). So that part is correct.

Let's recalculate with more precise numbers:

\( Q = 11.0\times 10^{-6}\space C \), \( q = 7.26\times 10^{-6}\space C \), so \( Q + q = 18.26\times 10^{-6}\space C \)

\( k = 8.988\times 10^{9}\space Nm^{2}/C^{2} \)

\( r = 0.117\space m \), \( r^{2}=0.117\times0.117 = 0.013689\space m^{2} \)

\( E=\frac{8.988\times 10^{9}\times18.26\times 10^{-6}}{0.013689} \)

First, compute the numerator: \( 8.988\times 18.26 = 8.988\times18 + 8.988\times0.26 = 161.784 + 2.33688 = 164.12088 \)

Then, \( 164.12088\times 10^{9 - 6}=164.12088\times 10^{3}=164120.88 \)

Now divide by 0.013689: \( 164120.88\div0.013689 = 164120.88\div0.013689 \approx 12000000 \) (since 0.013689 * 12000000 = 164268, which is very close to 164120.88, the slight difference is due to rounding in the multiplication of 8.988 and 18.26). So the electric field is approximately \( 1.20\times 10^{7}\space N/C \) or \( 12.0\space MN/C \)

Answer:

\( \boxed{1.20\times 10^{7}} \) (or \( 12.0\times 10^{6} \))