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A conducting sphere of radius 10 cm is charged with 10 μC. Another uncharged sphere of radius 20 cm in allowed to touch it for some time. If both the spheres are separated, then surface density of charges on the spheres will be in the ratio of
- a)1:4
- b)4:1
- c)1:2
- d)2:!
Correct answer is option 'D'. Can you explain this answer?
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Solution:
Given: Radius of conducting sphere, r1 = 10 cm
Charge on conducting sphere, q1 = 10 μC
Radius of uncharged sphere, r2 = 20 cm
Step 1: Calculation of Surface Charge Density of Charged sphere
Surface charge density, σ = q/A
where q is the charge and A is the surface area of the sphere.
Surface area of sphere, A = 4πr1^2
σ1 = q1/A
σ1 = (10 × 10^-6 C) / (4π × (10 × 10^-2 m)^2)
σ1 = 7.96 × 10^-5 C/m^2
Step 2: Calculation of Charge on Uncharged Sphere
Charge on uncharged sphere, q2 = (4π/3)ε0r2σ1
where ε0 is the permittivity of free space.
q2 = (4π/3)ε0r2σ1
q2 = (4π/3) × 8.85 × 10^-12 C^2/Nm^2 × (20 × 10^-2 m) × 7.96 × 10^-5 C/m^2
q2 = 5.59 × 10^-8 C
Step 3: Calculation of Surface Charge Density of Uncharged Sphere
Surface charge density of uncharged sphere, σ2 = q2/A
where A is the surface area of the sphere.
Surface area of sphere, A = 4πr2^2
σ2 = q2/A
σ2 = (5.59 × 10^-8 C) / (4π × (20 × 10^-2 m)^2)
σ2 = 2.79 × 10^-5 C/m^2
Step 4: Calculation of Ratio of Surface Charge Density
Ratio of surface charge density, σ1:σ2 = (7.96 × 10^-5 C/m^2) : (2.79 × 10^-5 C/m^2)
Ratio of surface charge density, σ1:σ2 = 2.85 : 1
Step 5: Final Answer
Hence, the surface density of charges on the spheres will be in the ratio of 2 : 1 (approximately), which is option (d).
Given: Radius of conducting sphere, r1 = 10 cm
Charge on conducting sphere, q1 = 10 μC
Radius of uncharged sphere, r2 = 20 cm
Step 1: Calculation of Surface Charge Density of Charged sphere
Surface charge density, σ = q/A
where q is the charge and A is the surface area of the sphere.
Surface area of sphere, A = 4πr1^2
σ1 = q1/A
σ1 = (10 × 10^-6 C) / (4π × (10 × 10^-2 m)^2)
σ1 = 7.96 × 10^-5 C/m^2
Step 2: Calculation of Charge on Uncharged Sphere
Charge on uncharged sphere, q2 = (4π/3)ε0r2σ1
where ε0 is the permittivity of free space.
q2 = (4π/3)ε0r2σ1
q2 = (4π/3) × 8.85 × 10^-12 C^2/Nm^2 × (20 × 10^-2 m) × 7.96 × 10^-5 C/m^2
q2 = 5.59 × 10^-8 C
Step 3: Calculation of Surface Charge Density of Uncharged Sphere
Surface charge density of uncharged sphere, σ2 = q2/A
where A is the surface area of the sphere.
Surface area of sphere, A = 4πr2^2
σ2 = q2/A
σ2 = (5.59 × 10^-8 C) / (4π × (20 × 10^-2 m)^2)
σ2 = 2.79 × 10^-5 C/m^2
Step 4: Calculation of Ratio of Surface Charge Density
Ratio of surface charge density, σ1:σ2 = (7.96 × 10^-5 C/m^2) : (2.79 × 10^-5 C/m^2)
Ratio of surface charge density, σ1:σ2 = 2.85 : 1
Step 5: Final Answer
Hence, the surface density of charges on the spheres will be in the ratio of 2 : 1 (approximately), which is option (d).
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A conducting sphere of radius 10 cm is charged with 10 μC. Another uncharged sphere of radius 20 cm in allowed to touch it for some time. If both the spheres are separated, then surface density of charges on the spheres will be in the ratio ofa)1:4b)4:1c)1:2d)2:!Correct answer is option 'D'. Can you explain this answer?
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