Cylindrical Capacitor Formula: Definition, Explanation

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A cylindrical capacitor is used for storing large amounts of electric current in a small space. It is made up of a hollow or a solid cylindrical conductor surrounded by a concentric hollow spherical cylinder. The cylindrical capacitors are used in electric motors, flour mills, electric juicers, and other electrical appliances. A capacitor is structured in such a way that it restricts the electric field lines within a small space. So, even if the field may have considerable strength, the potential difference between the two conductors of a capacitor is small. Though capacitors are grouped in several ways to get the desired capacitance, series and parallel are usual ways of grouping. The capacitance of a capacitor is measured in Farad (F). In this article, we will learn about the formula to find cylindrical capacitance and how to use it in solving problems.

Table of Content

Explanation of formula
Things to Remember
Sample Questions

Key Takeaways: Capacitance, radius, permittivity, torque, potential energy, potential difference, Coulomb, Farad, electrostatic force.

Explanation of Formula

[Click Here for Sample Questions]

A cylindrical capacitor is generally used for the storage of electric charge. The cylindrical capacitor possesses the shape of a cylinder with the inner radius as ‘a’ and the outer radius as ‘b.’

The Formula for the Capacitance of Cylindrical Capacitor is given below:

C = (2πε₀ x L) / ln(b/a)

The formula is also known as the cylindrical capacitor equation.

Here,

C is the capacitance of the cylinder

a is the inner radius of the cylinder

L is the length of the cylinder

b is the outer radius of the cylinder

ε0 is the permittivity of free space

Read more: The Parallel Plate Capacitor

Things to Remember

The potential due to a charge q at its location is not defined – it is infinite.
The space inside a conductor is shielded from outside electrical influences. If we put charges inside the space, the exterior of the conductor is not shielded from the fields by the inside charges.
The capacitor is so structured that it restricts the electric field lines within a small space. Therefore, although the field may have considerable strength, the potential difference between the two conductors of a capacitor is small. (Learn about Types of Capacitors)
A cylindrical capacitor is made up of a hollow or a solid cylindrical conductor surrounded by a concentric hollow spherical cylinder.
Capacitance of Cylindrical Capacitor is, C = (2πε₀ x L) / ln(b/a).

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Sample Questions

Ques. Can a capacitor be charged indefinitely? (2 marks)

Ans. When we charge a capacitor gradually, it means that the provided voltage is within its rated voltage. It will keep on assimilating the charge from the flow of current until it creates a potential equal and opposite to the supply voltage. After that, because of the capacitor's potential, the current flow can be stopped, and the capacitor charge will remain constant. If a larger voltage is supplied to the capacitor, the material present inside the capacitor will undergo a "breakdown," causing it to conduct between the terminals and hence causing the current to flow freely through it.

Ques. What is the reason behind the capacitor explosion? (2 marks)

Ans. The major reason behind the capacitor explosion is overvoltage. There are two conducting parts in a capacitor, including the insulation material between them. This layer of insulation has di-electric strength up to some certain limit and if any high voltage greater than rated is applied across the capacitor, its dielectric strength will instantly break and the capacitor will explode.

Ques. Why are many capacitors present in cylindrical shapes? (2 marks)

Ans. The general form of the capacitor is given to be as two conductive plates, that are separated by an insulating gap. There are several ways to manufacture the insulating and conductive components, with metal foils, metalized polymer films, electrolytes, and more. All of the above can be made in a flat rectangular form. Some capacitors are not cylindrical. They tend to be made by the stacking layers that alternate between insulating and conducting material. For example, we have multilayer ceramic chip capacitors.

Ques. Explain the reason for the cylindrical shape in motors? (2 marks)

Ans. The most efficient shape for heat dissipation is a round shape. This is because it uses the least amount of metal in the manufacturing process. The inner rotor has to be round to have the smallest distance between itself and the magnetic field of the stator winding. This is essential for maximum efficiency and this specific stator magnetic coupling rotates the armature.

Ques. What is a cylindrical capacitor made of? State few uses. (2 marks)

Ans. A cylindrical capacitor is made up of a cylindrical conductor surrounded by a concentric hollow spherical cylinder. The main use of cylindrical capacitors occurs in electric motors, flour mills, electric juicers, and other such electrical instruments.

Ques. Find the capacitance of a cylindrical capacitor filled with paper and formed by cylinders with inner radius 2 cm, and outer radius 5 cm. (2 marks)

Ans From the formula for the Capacitance of Cylindrical Capacitor, we substitute the permittivity, which equals to 3.85 for paper, thus:

C = (2πε₀x L) / ln (b/a)

C = 2 (3.14) (8.85 x 10^-12) (3.85) / ln (5/2)

C = 3.85 x 55.63 x 10^-12 / ln (2.5)

C = 538.22 x 10^-12 F

Ques. A cylindrical capacitor is made of two rings including an inner radius of 6 cm and an outer radius of 12 cm. Calculate the capacitance of the capacitor if its length is 16 cm. (3 marks)

Ans. We are given that,

Inner radius a = 6 cm

Length L = 16 cm

Outer radius b = 12 cm

Formula for the Capacitance of Cylindrical Capacitor:

C = (2πε₀ x L) / ln (b/a)

C = (2) (3.14) (8.85 x 10^-12) (16 x 10^-2) / ln (12/6)

C = 889.24 x 10^-14 / 0.301

C = 2.954 x 10^-11 F

Ques. The length of a cylindrical capacitor is 8 cm. It is made of two concentric rings including an inner radius of 3 cm and an outer radius of 6 cm. Calculate the capacitance of the capacitor. (3 marks)

Ans. We are given that,

Inner radius a = 3 cm

Length L = 8 cm

Outer radius b = 6 cm

Formula for the Capacitance of Cylindrical Capacitor:

C = (2πε₀ x L) / ln (b/a)

C = (2) (3.14) (8.85 x 10^-12) (8 x 10^-2) / ln (6/3)

C = 444.62 x 10^-14 / 0.301

C = 1.477 x 10^-11 F

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Cylindrical Capacitor Formula: Definition, Explanation

Explanation of Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

3.
Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

4.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

5.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

6.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

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Cylindrical Capacitor Formula: Definition, Explanation

Explanation of Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

3.Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

4.Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

5. If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

6.What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?

Comments

SUBSCRIBE TO OUR NEWS LETTER

3.
Two small identical metallic balls having charges \( q \) and \( -2q \) are kept far at a separation \( r \). They are brought in contact and then separated at distance \( \frac{r}{2} \). Compared to the initial force \( F \), they will now:

4.
Two thin lenses of focal length \( f_1 \) and \( f_2 \) are placed in contact with each other coaxially. Prove that the focal length \( f \) of the combination is given by \[ f = \frac{f_1 f_2}{f_1 + f_2}. \]

5.
If both the number of protons and the neutrons are conserved in each nuclear reaction, in what way is mass converted into energy (or vice versa) in a nuclear reaction? Explain.

6.
What is displacement current (\( i_d \))? Considering the case of charging of a capacitor, show that \( i_d = \varepsilon_0 \frac{d\Phi_E}{dt} \). What is the value of \( i_d \) for a conductor across which a constant voltage is applied?