Name: capacitance in spheres and cylinders
Uploaded: 2025-05-22T09:05:50.000Z
Duration: 1 min 26 s
Channel: JoVE

K12 Standards

Tennessee Academic Standards

High School

Physics

Energy

PHYS.PS3.13

Spherical and Cylindrical Capacitor

A spherical capacitor consists of two oppositely charged concentric spherical shells separated by an insulator. The inner shell radius is R1, and the outer shell radius is R2.
Considering a spherical Gaussian surface of radius r, the radially outward electric field can be expressed using the Gauss Law. The electric field is directly proportional to the charge enclosed and inversely proportional to the radius square.
Recall that potential difference can be derived from the electric field. Therefore, integrating the electric field along a radial path between the shells gives the potential difference for a spherical capacitor.
Now, the ratio of charge to the potential difference gives the capacitance for a spherical capacitor.
When the concentric spherical shells are replaced with concentric conducting cylinders, a cylindrical capacitor is formed.
Applying Gauss Law, the electric field directed radially outward from the common axis of the cylinder is calculated.
The potential difference calculated from the electric field can be applied to estimate the capacitance for a cylindrical capacitor.

spherical and cylindrical capacitor

A spherical capacitor consists of two concentric conducting spherical shells of radii R1 (inner shell) and R2 (outer shell). The shells have equal and opposite charges of +Q and &#8722;Q, respectively. For an isolated conducting spherical capacitor, the radius of the outer shell can be considered to be infinite.
Conventionally, considering the symmetry, the electric field between the concentric shells of a spherical capacitor is directed radially outward. The magnitude of the field, calculated by applying Gauss&#8217;s law over a spherical Gaussian surface of radius r concentric with the shells, is given by,
<img alt="Gauss's law equation for electric flux, integral form, physics concept diagram." src="/files/ftp_upload/13735/13735_Equation_PT_1.svg" style="text-align: center; height: 32px;" data-altupdatedat="1754904545000" data-alt="Gauss's law equation diagram showing electric flux, integral form, electrostatics concept." />
Substitution of the electric field into the electric field-capacitance relation gives the electric potential as,
<img alt="Electrostatics potential formula, V=Q/4&#960;&#949;&#8320;(1/R&#8321;-1/R&#8322;), equation for electric field analysis." src="/files/ftp_upload/13735/13735_Equation_PT_2.svg" style="text-align: center; height: 32px;" data-altupdatedat="1754904545000" data-alt="Electric potential equation \( V = \frac{Q}{4\pi\varepsilon_0} \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \)." />
However, since the radius of the second sphere is infinite, the potential is given by,
<img alt="Electrostatic potential formula, V=Q/4&#960;&#949;&#8320;(1/R&#8321;); static equilibrium, physics equation." src="/files/ftp_upload/13735/13735_Equation_PT_3.svg" style="text-align: center; height: 32px;" data-altupdatedat="1754904545000" data-alt="Electrostatics equation, Coulomb's law, V=Q/(4&#960;&#949;&#8320;(1/R&#8321;)), physics formula." />
Since, the ratio of charge to potential difference is the capacitance, the capacitance of an isolated conducting spherical capacitor is given by,
<img alt="Capacitance formula C=4&#960;&#949;&#8320;R&#8321;, mathematical equation for physics, static equilibrium concept." src="/files/ftp_upload/13735/13735_Equation_PT_4.svg" style="text-align: center; height: 14px;" data-altupdatedat="1754904545000" data-alt="Capacitance calculation formula, C=4&#960;&#949;&#8320;R&#8321;, highlighting electrostatic process." />
A cylindrical capacitor consists of two concentric conducting cylinders of length l and radii R1 (inner cylinder) and R2 (outer cylinder). The cylinders are given equal and opposite charges of +Q and -Q, respectively. Consider the calculation of the capacitance of a cylindrical capacitor of length 5 cm and radii 2 mm and 4 mm.
The known quantities are the capacitor&#8217;s length and inner and outer radii. The unknown quantity capacitance can be calculated using the known values.
The capacitance of a cylindrical capacitor is given by,
<img alt="Capacitance equation diagram, C=2&#960;&#949;&#8320;l/ln(R&#8322;/R&#8321;), relevant to electric field studies." src="/files/ftp_upload/13735/13735_Equation_PT_5.svg" style="text-align: center; height: 32px;" data-altupdatedat="1754904545000" data-alt="Capacitance formula C=(2&#960;&#949;&#8320;l)/ln(R&#8322;/R&#8321;); Equation for cylindrical capacitor analysis." />
When the known values are substituted into the above equation, the calculated capacitance value is 4.02 pF.

A spherical capacitor consists of two concentric conducting spherical shells of radii R1&#160;(inner shell) and R2&#160;(outer shell). The shells ...

Education

JoVE Core

Unit 1

Capacitance

KEY TERMS AND CONCEPTS

capacitance in spheres and cylinders

A spherical capacitor uses two concentric conducting spherical shells. The inner shell has radius R1, and the outer shell has radius R2. The shells carry equal and opposite charges, +Q and −Q.

For an isolated conducting spherical capacitor, the outer shell can be treated as if it were at infinity. By symmetry, the electric field between the shells points radially outward. Using Gauss’s law on a spherical Gaussian surface of radius r gives the field magnitude between the shells.

That electric field can be used to find the electric potential. When the outer sphere is at infinity, the potential takes a simpler form. The capacitance follows from the ratio of charge to potential difference, so an isolated spherical capacitor has a defined capacitance expression.

A cylindrical capacitor uses two concentric conducting cylinders. The inner cylinder has radius R1, the outer cylinder has radius R2, and the cylinders have length l. They also carry equal and opposite charges, +Q and −Q.

The transcript includes a sample calculation for a cylindrical capacitor with a length of 5 cm and radii of 2 mm and 4 mm. These values are substituted into the capacitance equation for a cylindrical capacitor. The result is a capacitance of 4.02 pF.

Capacitance in Spheres and Cylinders

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A spherical capacitor uses two concentric conducting spherical shells. The inner shell has radius R1, and the outer shell has radius R2. The shells carry equal and opposite charges, +Q and −Q. For an isolated conducting spherical capacitor, the outer shell can be treated as if it were at infinity. By symmetry, the electric field between the shells points radially outward. Using Gauss’s law on a spherical Gaussian surface of radius r gives the field magnitude between the shells. That electric...

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key terms and concepts

Learn how spherical and cylindrical capacitors are built, how Gauss’s law helps find the field, and how a 5 cm cylindrical capacitor gives 4.02 pF.

Capacitance in Spheres and Cylinders | JoVE K12

physics

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Virginia  Standards of Learning

HIgh School

Physical Science

PS.9

PS.9.a

PH.8

PH.8.a

Electrical Current

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Resistance

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Ohm's Law

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PHYS.PS3.11

Kirchhoff's Rules

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PHYS.PS3.12

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PHYS.PS3.15

Capacitance in Spheres and Cylinders

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