Q- The radii of spheres forming a spherical condenser are 0.5 and 0.6 meter. if a medium having dielectric constant 6 is completely filled in between the capacity of the condenser will be.

To find the capacitance of a spherical condenser with given radii and a dielectric medium, we can use the formula for the capacitance of a spherical capacitor. The formula is given by:

Capacitance Formula

The capacitance \( C \) of a spherical capacitor is expressed as:

C = \frac{4 \pi \varepsilon_0 \varepsilon_r}{\frac{1}{r_1} - \frac{1}{r_2}}

Where:

• C = capacitance in farads (F)
• \(\varepsilon_0\) = permittivity of free space, approximately \(8.85 \times 10^{-12} \, \text{F/m}\)
• \(\varepsilon_r\) = relative permittivity (dielectric constant) of the medium
• r1 = radius of the inner sphere
• r2 = radius of the outer sphere

Given Values

From your question, we have:

• Inner radius \( r_1 = 0.5 \, \text{m} \)
• Outer radius \( r_2 = 0.6 \, \text{m} \)
• Dielectric constant \( \varepsilon_r = 6 \)

Calculating Capacitance

Now, we can substitute these values into the formula. First, we need to calculate the difference in the reciprocals of the radii:

\(\frac{1}{r_1} - \frac{1}{r_2} = \frac{1}{0.5} - \frac{1}{0.6} = 2 - \frac{5}{6} = 2 - 0.8333 = 1.1667 \, \text{m}^{-1}\)

Next, we can substitute this into the capacitance formula:

C = \frac{4 \pi (8.85 \times 10^{-12}) (6)}{1.1667}

Calculating the numerator:

4 \pi (8.85 \times 10^{-12}) (6) \approx 6.69 \times 10^{-10} \, \text{F}

Now, dividing by \( 1.1667 \):

C \approx \frac{6.69 \times 10^{-10}}{1.1667} \approx 5.73 \times 10^{-10} \, \text{F}

Final Result

Thus, the capacitance of the spherical condenser filled with a dielectric medium of constant 6 is approximately:

C ≈ 5.73 \times 10^{-10} \, \text{F} \, (or \, 573 \, \text{pF})

This value indicates how much electric charge the capacitor can store per unit voltage applied across it, and it reflects the influence of the dielectric material in enhancing the capacitance compared to a vacuum or air-filled capacitor.