Question

A Van de Graaff generator has a spherical conductor with a radius of \(25.0 \mathrm{~cm} .\) It can produce a maximum electric field of \(2.00 \cdot 10^{6} \mathrm{~V} / \mathrm{m}\). What are the maximum voltage and charge that it can hold?

Short answer

Answer: The maximum charge and voltage that the Van de Graaff generator can hold are approximately \(4.39 \cdot 10^{-5} \mathrm{C}\) and \(7.92 \cdot 10^{5} \mathrm{V}\), respectively.

Step-by-step solution

Identify the given values

First, we can identify the given values in the problem: 1. Radius (r) of the spherical conductor: 25.0 cm or 0.25 m. 2. Maximum electric field (E): \(2.00 \cdot 10^{6} \mathrm{~V/m}\).

Use the electric field formula for a spherical conductor

For a spherical conductor, the electric field is given by the formula: \(E = \frac{Q}{4 \pi \epsilon_0 r^2}\) where Q is the charge, r is the radius of the conductor, and \(\epsilon_0\) is the vacuum permittivity constant (\(\approx 8.85 \cdot 10^{-12} \mathrm{C^{2}/N.m^{2}}\)). We will now solve for the charge Q: \(Q = 4 \pi \epsilon_0 r^2 E\)

Calculate the charge

Plugging in the given values and the vacuum permittivity constant, we can now calculate the maximum charge that the Van de Graaff generator can hold: \(Q = 4 \pi (8.85 \cdot 10^{-12} \mathrm{C^{2}/N.m^{2}}) (0.25 \mathrm{m})^2 (2.00 \cdot 10^{6} \mathrm{V/m})\) \(Q = 4.39 \cdot 10^{-5} \mathrm{C}\) The maximum charge that the Van de Graaff generator can hold is approximately \(4.39 \cdot 10^{-5} \mathrm{C}\).

Use the voltage formula for a spherical conductor

Now we need to find the maximum voltage that the generator can hold. For a spherical conductor, the voltage V is given by the formula: \(V = \frac{Q}{4 \pi \epsilon_0 r}\) We already know the value of Q from step 3. Now we can plug in the values to find the maximum voltage:

Calculate the voltage

\(V = \frac{4.39 \cdot 10^{-5} \mathrm{C}}{4 \pi (8.85 \cdot 10^{-12} \mathrm{C^{2}/N.m^{2}})(0.25 \mathrm{m})}\) \(V = 7.92 \cdot 10^{5} \mathrm{V}\) The maximum voltage that the Van de Graaff generator can hold is approximately \(7.92 \cdot 10^{5} \mathrm{V}\). Therefore, the maximum charge and voltage that the Van de Graaff generator can hold are \(4.39 \cdot 10^{-5} \mathrm{C}\) and \(7.92 \cdot 10^{5} \mathrm{V}\), respectively.

Key concepts

Electric Field

The electric field is a fundamental concept in electromagnetism representing the force that a charged particle exerts on other charges in the space around it. It's often visualized as lines emanating from a charge or an object that has a net charge. The density of these lines indicates the strength of the electric field at any given point.

For a spherical conductor, like the sphere of a Van de Graaff generator, the electric field at the surface is particularly simple to describe: it's directly proportional to the charge on the sphere and inversely proportional to the square of the radius of the sphere. In mathematical terms, for a spherical conductor, the electric field (E) is described by the equation: \[E = \frac{Q}{4 \pi \epsilon_0 r^2}\]
where \(Q\) is the charge on the sphere, \(r\) is the radius, and \(\epsilon_0\) is the permittivity of free space. This relationship is crucial in understanding how the Van de Graaff generator works and dictates the maximum charge it can hold before breakdown occurs.

Spherical Conductor

A spherical conductor, such as the dome of a Van de Graaff generator, is an excellent illustration of electrostatic principles. Unlike irregular objects, a sphere's symmetry means that any excess charge is distributed evenly over its surface, resulting in a uniform charge density.

This uniform distribution is due to the repulsive forces among like charges and their tendency to be as far apart as possible, a phenomenon known as electrostatic equilibrium. As a result of this effect, the electric field inside a charged spherical conductor is zero. This is a crucial concept known as electrostatic shielding. It is this uniformly distributed charge that allows the Van de Graaff generator to hold a high voltage without discharging prematurely.

Voltage

Voltage, often referred to as electric potential difference, is a measure of the potential energy per unit charge at a point in an electric field. It represents the work done to move a charge between two points in the field against the electric forces.

In the context of a spherical conductor like the one on the Van de Graaff generator, the voltage at the surface is the amount of work required to bring a unit charge from infinity to the conductor's surface. Given the symmetry of the sphere, this value is consistent regardless of the path taken. The formula to calculate the voltage (V) for a spherical conductor is: \[V = \frac{Q}{4 \pi \epsilon_0 r}\]
Understanding voltage is key to comprehending how the Van de Graaff generator functions. High voltage allows the generator to create a substantial electric field and the potential to do work, such as accelerating charged particles or creating sparks.

Charge

Charge is one of the most fundamental properties of matter, and it comes in two types: positive and negative. In an isolated system, charge can neither be created nor destroyed, only transferred from one part to another. This is known as the conservation of charge.

In the realm of electrostatics, and specifically for devices like the Van de Graaff generator, charge accumulation on the spherical conductor is limited by the maximum electric field that it can sustain. This is because when the electric field exceeds a certain threshold, the surrounding air or material begins to ionize, leading to a discharge. The formula used to calculate the charge on the Van de Graaff generator's spherical conductor can be represented as: \[Q = 4 \pi \epsilon_0 r^2 E\]
Understanding how charge behaves, especially in equilibrium, allows us to manipulate it for various applications, such as in creating high-energy particles or in the study of electrostatic phenomena.