Advertisements
Advertisements
प्रश्न
A capacitor is formed by two square metal-plates of edge a, separated by a distance d. Dielectrics of dielectric constant K1 and K2 are filled in the gap as shown in figure . Find the capacitance.
Advertisements
उत्तर
Let us consider an elemental capacitor of width dx at a distance x from the left end of the capacitor. It has two capacitive elements of dielectric constants K1 and K2 with plate separations (x tan θ) and (d − x tan θ) in series, respectively. The areas of the plates of the capacitors are adx.
The capacitances of the capacitive elements of the elemental capacitor are :
`dC_1 = (∈_0K_2(adx))/(x tan θ) , dC_2 = (∈_0K_1(adx))/(d - x tan θ)`
The net capacitance of the elemental capacitor is given by
`1/(dC)= 1/(dC_1) + 1/(dC_2)`
`1/(dC) = (x tanθ)/(∈_0K_2(adx)) + (d-x tanθ)/(∈_0K_1(adx))`
`⇒ dC = (∈_0K_1K_2(adx))/(K_1 x tanθ + K_2(d-x tanθ)`
Thus, integrating the above expression to calculate the net capacitance
`C = ∫_0^a dC = ∫_0^a (∈_0K_1K_2adx)/(K_1x tanθ+ K_2(d-x tanθ)`
`⇒ C = ∈_0K_1K_2a ∫_0^a (dx)/(K_2d + x tanθ(K_1 - K_2)`
`⇒ C = (∈_0K_1K_2a)/(tanθ(K_1 - K_2))[log_e[K_2d + x tanθ(K_1 - K_2) ]]_0^a`
`⇒ C = (∈_0K_1K_2a)/(tan θ(K_1 - K_2)) [ log_e[K_2d + a tan θ(K_1 - K_2)] - log_e K_2d]`
As we know that `tan θ = d/a` substituting in the expression for capacitance C.
Now,
`⇒ C = (∈_0K_1K_2a)/(d/a xx (K_1 - K_2) )[ log_e[K_2d + a xx d/a(K_1 - K_2)] - log_e K_2d]`
`⇒ C = (∈_0K_1K_2a)/(d/a xx (K_1 - K_2) )[ log_e K_1d - log_e K_2d ]`
`⇒ C = (∈_0K_1K_2a^2)/(d(K_1 - K_2)) [log_e (K_1/K_2)]`
APPEARS IN
संबंधित प्रश्न
A bulb is connected in series with a variable capacitor and an AC source as shown. What happens to the brightness of the bulb when the key is plugged in and capacitance of the capacitor is gradually reduced?
(i) Find equivalent capacitance between A and B in the combination given below. Each capacitor is of 2 µF capacitance.
(ii) If a dc source of 7 V is connected across AB, how much charge is drawn from the source and what is the energy stored in the network?
A capacitor of unknown capacitance is connected across a battery of V volts. The charge stored in it is 360 μC. When potential across the capacitor is reduced by 120 V, the charge stored in it becomes 120 μC.
Calculate:
(i) The potential V and the unknown capacitance C.
(ii) What will be the charge stored in the capacitor, if the voltage applied had increased by 120 V?
A capacitor of unknown capacitance is connected across a battery of V volts. The charge stored in it is 300 μC. When potential across the capacitor is reduced by 100 V, the charge stored in it becomes 100 μC. Calculate The potential V and the unknown capacitance. What will be the charge stored in the capacitor if the voltage applied had increased by 100 V?
Three identical capacitors C1, C2 and C3 of capacitance 6 μF each are connected to a 12 V battery as shown.
Find
(i) charge on each capacitor
(ii) equivalent capacitance of the network
(iii) energy stored in the network of capacitors
A capacitor of capacitance C is charged to a potential V. The flux of the electric field through a closed surface enclosing the capacitor is
A finite ladder is constructed by connecting several sections of 2 µF, 4 µF capacitor combinations as shown in the figure. It is terminated by a capacitor of capacitance C. What value should be chosen for C, such that the equivalent capacitance of the ladder between the points A and B becomes independent of the number of sections in between?
Consider the situation shown in the figure. The switch S is open for a long time and then closed. (a) Find the charge flown through the battery when the switch S is closed. (b) Find the work done by the battery.(c) Find the change in energy stored in the capacitors.(d) Find the heat developed in the system.
The separation between the plates of a parallel-plate capacitor is 0⋅500 cm and its plate area is 100 cm2. A 0⋅400 cm thick metal plate is inserted into the gap with its faces parallel to the plates. Show that the capacitance of the assembly is independent of the position of the metal plate within the gap and find its value.
Consider an assembly of three conducting concentric spherical shell of radii a, b and c as shown in figure Find the capacitance of the assembly between the points Aand B.
Figure shows two parallel plate capacitors with fixed plates and connected to two batteries. The separation between the plates is the same for the two capacitors. The plates are rectangular in shape with width b and lengths l1 and l2. The left half of the dielectric slab has a dielectric constant K1 and the right half K2. Neglecting any friction, find the ration of the emf of the left battery to that of the right battery for which the dielectric slab may remain in equilibrium.
Obtain an expression for equivalent capacitance when three capacitors C1, C2 and C3 are connected in series.
Three capacitors are connected in a triangle as shown in the figure. The equivalent capacitance between points A and C is ______.
Explain in detail the effect of a dielectric placed in a parallel plate capacitor.
Two spherical conductors A and B of radii a and b(b > a) are placed concentrically in the air. B is given a charge +Q and A is earthed. The equivalent capacitance of the system is ______.
The radius of a sphere of capacity 1 microfarad in the air is ______
Can the potential function have a maximum or minimum in free space?
A parallel plate capacitor (A) of capacitance C is charged by a battery to voltage V. The battery is disconnected and an uncharged capacitor (B) of capacitance 2C is connected across A. Find the ratio of final charges on A and B.
