Spherical capacitor formed by spheres of radii

A spherical capacitor consists of two spheres with radii R1=4cm and R2=6cm, which were charged to a voltage of 1 kV and then disconnected from the source. It is assumed that a point is located at a distance of 5 cm from the center of the spheres. It is required to determine how much the potential of this point will change if the radius of the outer sphere increases to R3 = 10 cm, provided that the outer sphere is grounded.

First you need to determine the capacitance of the capacitor. The capacitance of a spherical capacitor can be found using the formula:

C = 4πε₀ ((R₁R₂)/(R₂-R₁))

where ε₀ is the electrical constant, R₁ and R₂ are the radii of the inner and outer spheres, respectively.

Substituting the known values, we get:

C = 4πε₀ ((4cm×6cm)/(6cm-4cm)) = 1.69·10⁻¹⁰ F

The charge on each sphere can be found using the formula:

Q = CU

where U is the voltage across the capacitor.

Substituting the known values, we get:

Q₁ = C·U = 1,69·10⁻¹⁰·1000 = 1,69·10⁻⁷ Кл Q₂ = C·U = 1,69·10⁻¹⁰·1000 = 1,69·10⁻⁷ Кл

The charge on the outer sphere is zero since it is grounded.

To determine the potential of a point at a distance of 5 cm from the center of the spheres, you must use the formula for the potential of a point charge:

V = kQ/r

where k is the proportionality coefficient, r is the distance from the point to the charge.

The potential of a point at a distance of 5 cm from the center of the spheres to the charges is in a capacitor with capacitance C and charge Q₁+Q₂. Thus, the potential of a point can be found as the sum of the potentials created by the charges on each sphere and the potential created by the external grounded sphere. According to the principle of superposition:

V = k(Q₁+Q₂)/r₁ + k(0)/r₂

where r₁ is the distance from the point to the center of the inner sphere, r₂ is the distance from the point to the center of the outer sphere.

Substituting the known values, we get:

V = k(1.69·10⁻⁷)/(0.05) + k(0)/(0.1) = 2.71 V

Now it is necessary to find the capacitance of the capacitor after increasing the radius of the outer sphere to R3=10cm. The capacitance of a spherical capacitor can be found using the formula:

C' = 4πε₀ ((R₁R₃)/(R₃-R₁))

Substituting the known values, we get:

C' = 4πε₀ ((4cm×10cm)/(10cm-4cm)) = 3.38·10⁻¹⁰ F

The charge on each sphere will remain unchanged since they are disconnected from the source. Consequently, the charge on the inner sphere will remain equal to Q₁=1.69·10⁻⁷ C, and the charge on the outer sphere will remain equal to zero.

To determine the new potential of a point at a distance of 5 cm from the center of the spheres, you must use the same formula:

V' = k(Q₁+Q₂)/r₁' + k(0)/r₂'

where r₁' is the new distance from the point to the center of the inner sphere, r₂' is the new distance from the point to the center of the outer sphere.

The new distance from the point to the center of the inner sphere can be found through the Pythagorean theorem:

r₁' = √(r₁² + (R₃-R₂)²) = √(0.05² + (10cm-6cm)²) = 0.61 cm

The new distance from the point to the center of the outer sphere can also be found through the Pythagorean theorem:

r₂' = √(r₂² + (R₃-R₂)²) = √(0.1² + (10cm-6cm)²) = 0.77 cm

Substituting the known values, we get:

V' = k(1.69·10⁻⁷)/(0.61) + k(0)/(0.77) = 2.15 V

The change in the potential of a point can be found as the difference between the new and old potentials:

ΔV = V' - V = 2,15 В - 2,71 В = -0,56 В

Thus, the potential of a point at a distance of 5 cm from the center of the spheres will decrease by 0.56 V when the radius of the outer sphere is increased to 10 cm and this sphere is grounded.

Product Description: Spherical Capacitor

This digital product is a description of a spherical capacitor formed by two spheres with radii:

• R1=4 cm
• R2=6 cm

The capacitor is charged to a voltage of 1 kV and disconnected from the source. The distance from the center of the spheres to the point at which the potential is determined is 5 cm. The outer sphere is grounded.

In this description you will find a detailed solution to problem 30346, which includes a brief recording of the conditions, formulas and laws used in the solution, derivation of the calculation formula and the answer to the problem.

If you have any questions about solving the problem, do not hesitate to contact us. We are always happy to help!

Product Description: Spherical Capacitor

This product is a description of a spherical capacitor formed by two spheres with radii R1=4cm and R2=6cm. The capacitor is charged to a voltage of 1 kV and disconnected from the source. The distance from the center of the spheres to the point at which the potential is determined is 5 cm. The outer sphere is grounded.

In this description you will find a detailed solution to problem 30346, which consists in determining the change in the potential of a point when the radius of the outer sphere increases to R3 = 10 cm, provided that the outer sphere is grounded. The solution uses the appropriate formulas and laws, provides calculations, and obtains an answer to the problem.

If you have any questions about solving the problem or about spherical capacitors in general, do not hesitate to contact us. We are always happy to help!

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A spherical capacitor, formed by two spheres with radii R1=4cm and R2=6cm, is designed to store an electric charge. The capacitor is charged to a voltage of 1 kV and disconnected from the source.

To solve the problem, we are given that at a distance of 5 cm from the center of the spheres there is a point at which we need to determine the change in potential if the radius of the outer sphere increases to R3 = 10 cm. The outer sphere is grounded.

To calculate the change in potential at a point at a distance of 5 cm from the center of the sphere, you can use Coulomb’s law, which states that the electric field E at a point located at a distance r from the center of a charged sphere with charge Q of radius R is equal to: E = Q/(4πε0r^2)

Here ε0 is the dielectric constant.

To calculate the change in potential at a point, you can use the formula: ΔV = - ∫E dl

Here the integral is taken along any path connecting the starting and ending points.

You can also use the formula to calculate the charge on spheres: Q = 4πε0R·ΔV

Here R is the radius of the sphere on which the charge is calculated.

Solution tasks:

Initial charge on the capacitor: Q1 = C U = (4πε0R1R2)/(R2-R1) U = (4πε0 4cm 6cm)/(6cm-4cm) 1000V = 100πε0μC

Charge on the outer sphere after increasing the radius: Q3 = 4πε0R3 ΔV

Change in potential at a point 5 cm from the center of the sphere: ΔV = - ∫E dl

To calculate the field E at a distance of 5 cm from the center of the sphere, you can use the formula: E = Q/(4πε0r^2)

To calculate the charge on the outer sphere, you can use the law of conservation of charge: Q1 + Q2 = Q3

Then the charge on the inner sphere is: Q2 = Q3 - Q1 = 4πε0(R3-R1)(R3+R1)/(R3-R1) ΔV = 4πε0(R3+R1) ΔV

Thus, the total change in potential at a point at a distance of 5 cm from the center of the sphere with an increase in the radius of the outer sphere from R2 to R3 will be equal to: ΔV = - ∫E dl = - E 2πr = - Q2/(4πε0r) = -(R3+R1) ΔV/r

Substituting numerical values, we get: ΔV = - (10cm+4cm) 1000V/5cm = - 2800V

Answer: The change in the potential of a point located at a distance of 5 cm from the center of the spheres, with an increase in the radius of the outer sphere from R2 = 6 cm to R3 = 10 cm, will be -2800 V.

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