spherical capacitor formula capacitance derivation conductor class 12

find spherical capacitor formula capacitance derivation conductor class 12 derive ?

Capacitance

When a conductor is given a charge Q, it acquires a potential V which is proportional to the charge given to it, i.e.

where C is a constant of proportionality and is called the capacitance which is defined as the amount of charge in coulomb necessary to increase the potential of a conductor by 1 volt. The SI unit of capacitance is farad (symbol F)

The farad is a large unit. More practical units are microfarad (μF) and picofarad (pF).

Energy of a Charged Conductor

A charged conductor has electric field in the region around it. If additional similar charge is given to the conductor, work has to be done against the electrical repulsive force. This work is stored in the form of potential energy which resides in the electric field. If a charge Q is given to a conductor of capacitance C, the potential energy in its electric field is given by

Capacitance of a Single Spherical Conductor

Consider a spherical conductor of radius r having a charge Q. Since the electric field is normal to the surface of the sphere, the lines of force appear to originate from its centre, i.e. the charge Q may be supposed to be concentrated at the centre. Therefore, the potential is given by

Since C = Q/V , the capacitance of the sphere is given by

C = 4πε₀r

Thus, the greater the radius of the sphere, the higher is its capacitance.

Capacitors Any isolated system of two conducting bodies, of any shape and size, separated by a distance is called a capacitor. If two conductors, carrying equal and opposite charge Q have a potential difference V between them,

then Q = CV

where C is the capacitance of the capacitor and its value depends on the size, the shape, the separation between the conductors and the nature of the medium between them. If C0 is the capacitance of the capacitor when the medium is air (or vacuum) and Cm its capacitance when the medium is a dielectric other than air, then the dielectric constant of the medium is given by

Expressions for Capacitance

1. Parallel Plate Capacitor

The capacitance of a parallel plate capacitor is given by

where A is the area of each plate and d is the distance between them. K is dielectric constant of the material between the plates. For air or vacuum, K = 1.

2. Spherical Capacitor

A spherical capacitor consists of a solid charged sphere of radius a surrounded by a concentric hollow sphere of radius b. Its capacitance is given by

3. Cylindrical Capacitor

A cylindrical capacitor consists of two co-axial cylinders and its capacitance is given by

where l is the length of each cylinder and a and b are the radii of the inner and outer cylinders.

4. If the space between the plates of a parallel plate capacitor is filled with two media of thicknesses d1 and d2 having dielectric constants K1 and K2, then the capacitance of the capacitor is given by

Capacitors in Parallel and Series

In parallel arrangement of capacitors, the potential difference across individual capacitors is the same and the total charge is shared by them in the ratio of their capacitances.

In series arrangement of capacitors, the charge on each capacitor is the same and the total potential difference is shared by them in the inverse ratio of their capacitances.

Therefore, the effective capacitance of the combination is given by

Energy Stored in a Capacitor

As in the case of a charged conductor, the energy stored in a capacitor is given by

where Q = charge on each plate of the capacitor, V = potential difference between plates and C = capacitance of the capacitor. This potential energy resides in the electric field in the medium between the plates

Loss of Energy on Sharing Charges

If two charged bodies carrying charges Q1 and Q2 and having capacitances C1 and C2 are connected with each other, then their common potential after the sharing of charges is given by

where V1 and V2 are the initial potentials of the charged bodies. The loss of energy is given by

Force between Plates of a Parallel Plate Capacitor

The plates of a capacitor carry equal and opposite charges. Therefore, they exert an attractive force on each other which is given by

The force per unit area of the plates is

where σ is the charge per unit area.

Wheatstone’s Bridge of Capacitors

In the circuit shown in Fig. below (a) in below, the network of capacitors form a wheatstone’s Bridge. The bridge is balanced if the values of C1, C2, C3 and C4 satisfy the condition

For networks which satisfy this condition, the equivalent capacitance can be found.

note : For an unbalanced Wheelstone’s bridge or for any other more complicated combinations of capacitors, it is not easy to find the equivalent capacitance using the formulae for series and parallel combinations.

For such cases, we should use the following procedure:

(1) Connect an imaginary battery between the points across which the equivalent capacitance is to be found.

(2) Send a positive charge + Q from the positive terminal of the battery and equal negative charge –Q from the negative terminal.

(3) Write the charges on each capacitor plate using the principle of charge conservation. i.e., charges on the two plates must be equal and opposite. Let Q1,Q2, … etc. be the charges on the capacitors in the network and V1, V2,… etc. be the respective potential differences.

(4) Use Q = CV for each capacitor. Eliminate Q1,Q2, … etc. and V1, V2,… etc. to obtain the equivalent capacitance Ceq = Q/E  , where E is the voltage of the battery

Charging and Discharging of a Capacitor through a Resistance

Consider a capacitor of capacitance C connected in series to a resistor R and a battery of emf E and negligible internal resistance through a two-way key as shown in Fig. below.

Growth of Charge The battery is introduced in the circuit by connecting terminals 1 and 2 of the key. Initially, i.e. when t = 0, the charge on the capacitor plates is zero. As time passes, the charge flows into the capacitor plates

and the potential difference q/C (q is the charge at any time t) between the plates rises. The charge on the plates of the capacitor rises till the potential difference between the plates becomes E. The maximum charge collected is q0 = CE. At this stage the current in the circuit becomes zero. The growth of charge on the capacitor plates as a function of time is given by

It is clear that the charge rises exponentially to a steady state maximum value q0 as shown in Fig. below (a).

Time Constant The product RC has the dimensions of time. If C is in farad and R in ohm, the product CR will be in seconds. Writing RC = τ in Eq. (1) we have

Thus the time constant τ of a CR circuit may be defined as the time during which the charge on the capacitor grows from zero to 0.63 of its maximum value q0. Whether the charge grows quickly or slowly depends on the value of the time constant, i.e. on the values of R and C. If the product CR (i.e. the time constant) is very small, the charge grows quickly. The behavior of current as a function of time t is given by

where I₀ = E/R is the maximum current. Therefore, the current decreases exponentially from its maximum value I₀ to zero as shown in Fig. above (b). Decay of Charge When the charge has attained a steady value EC, the battery is short-circuited by connecting the terminals 1 and 3 of the key K. In such a situation the capacitor starts discharging through the resistor, i.e. the charge on the capacitor starts flowing back through the resistor. The direction of the current is, therefore, reversed. The decay of charge with time is given by

Figure below shows the decay of charge with time. At time t = τ, q = q₀ e^–1 = 0.368 q₀. Thus in a time t = τ, the time constant, the charge on the capacitor decays to 0.368 of its initial value q₀. So the charge decays exponentially with time t. Whether the charge decays slowly or quickly depends on the value of the time constant RC.