1.A spherical condenser has inner and outer spheres of radii a and b respectively. The space between the two is filled air. The difference between the capacities of the two condensers formed when outer sphere is earthed and when inner sphere is earthed will be.........? Explain.
2. Charged pendulums are in equilibrium, making angle A, this arrangement is taken to rotating satellite, what will be the new value of A?
Thanks
1. To find the difference between the capacities of the two condensers, we need to first understand the concept of capacitance in a spherical condenser.
The capacitance (C) of a spherical condenser is given by the formula:
C = 4πε₀(ab) / (b - a),
where a is the radius of the inner sphere, b is the radius of the outer sphere, and ε₀ is the permittivity of free space.
Now, let's consider the two scenarios:
a) When the outer sphere is earthed:
In this case, the outer sphere becomes a reference point, and the inner sphere will have a potential difference with respect to the outer sphere.
The capacitance of the condenser when the outer sphere is earthed is given by:
C₁ = 4πε₀ab / (b - a).
b) When the inner sphere is earthed:
In this case, the inner sphere becomes a reference point, and the outer sphere will have a potential difference with respect to the inner sphere.
The capacitance of the condenser when the inner sphere is earthed is given by:
C₂ = 4πε₀ab / (a - b).
To find the difference between the capacities of the two condensers, we need to subtract the second capacitance from the first:
Difference = C₁ - C₂
= 4πε₀ab / (b - a) - 4πε₀ab / (a - b)
= [4πε₀ab(a - b) - 4πε₀ab(b - a)] / (b - a)(a - b)
= 0.
Hence, the difference between the capacities of the two condensers formed when the outer sphere is earthed and when the inner sphere is earthed is zero. This means the capacities are equal in both cases.
1. To find the difference between the capacities of the two condensers, we need to understand the concept of capacitance.
The capacitance of a capacitor is defined as the ratio of the charge stored on one of its plates to the potential difference across the plates, i.e., C = Q/V.
In the given scenario, we have a spherical condenser with inner and outer spheres of radii a and b, respectively. The space between the two spheres is filled with air or any other dielectric material.
When the outer sphere is earthed, the inner sphere still has a potential difference with respect to the earth. So, the capacity of the condenser formed is C1.
Similarly, when the inner sphere is earthed, the outer sphere still has a potential difference with respect to the earth. The capacity of the condenser formed in this case is C2.
The difference in capacities can be calculated using the formula:
ΔC = C2 - C1
Now, the capacitance of a spherical condenser is given by the formula:
C = 4πε₀(ab)/(b-a)
Substituting in the radius values, we have:
C1 = 4πε₀(ab)/(b-a)
C2 = 4πε₀(ab)/(a-b)
Calculating C2 - C1, we get:
ΔC = C2 - C1 = 4πε₀(ab)/(a-b) - 4πε₀(ab)/(b-a)
ΔC = 4πε₀(ab)[(a+b)/(a-b)] - 4πε₀(ab)[(b+a)/(b-a)]
ΔC = 8πε₀(ab)(a+b)/(a²-b²)
So, the difference between the capacities of the two condensers formed when the outer sphere is earthed and when the inner sphere is earthed is 8πε₀(ab)(a+b)/(a²-b²).
2. When a charged pendulum is in equilibrium, it means that the electrical force acting on the pendulum's charge is balanced by the gravitational force, resulting in a stable position.
When this arrangement is taken to a rotating satellite, the gravitational force acting on the pendulum will remain the same because the gravitational field strength near the surface of a satellite is similar to that on the Earth's surface.
However, the centrifugal force due to rotation will come into play. The centrifugal force depends on the angular velocity of the satellite and the radius of the pendulum's rotation.
The new value of angle A can be calculated using the equation:
tan(A) = (gravitational force)/(centrifugal force)
Since the gravitational force remains the same, the change will occur in the centrifugal force. As the satellite rotates faster, the centrifugal force will increase, leading to a larger angle A.
Therefore, the new value of angle A will be larger than the initial angle when the arrangement is taken to a rotating satellite.
There is no difference. THe capacitance is determined by the voltage difference between the spheres, and the two radii.
http://www.mwit.ac.th/~physicslab/hbase/electric/capsph.html
On earth, the charges repel, and the forces pulling the pendulum down is gravity. I orbit, gravity is equalled by centripetal force, giving the effect of no gravity, so the pendulums will be at 180 degrees.