A Van de Graaff generator (Fig. 23–58) can develop a very large potential difference, even millions of volts. Electrons are pulled off the belt by the high voltage pointed electrode (positive) at A, leaving the belt positively charged. (Recall Example 23–5 where we saw that near sharp points the electric field is high and ionization can occur.) The belt carries the positive charge up inside the spherical shell where electrons from the large conducting sphere are attracted over to the pointed conductor at B, leaving the outer surface of the conducting sphere positively charged. As more charge is brought up, the sphere reaches extremely high voltage. Consider a Van de Graaff generator with a sphere of radius 0.20 m. What is the electric potential on the surface of the sphere when electrical breakdown occurs ( E = 3 x 10⁶ V/m) ? Assume V = 0 at r = ∞.

Verified step by step guidance

1. Step 1: Understand the relationship between the electric field (E) and the electric potential (V). The electric field near the surface of a charged sphere is related to the potential by the formula: $E=\frac{V}{r}$, where E is the electric field, V is the potential, and r is the radius of the sphere.
2. Step 2: Rearrange the formula to solve for the electric potential (V). Using the formula from Step 1, we can write: $V=E\times r$. This equation will allow us to calculate the potential on the surface of the sphere.
3. Step 3: Substitute the given values into the formula. The problem states that the electric field at breakdown is $3\; \times \; 106V/m$ and the radius of the sphere is $0.20m$. Substitute these values into the equation: $V=\; (3\; \times \; 106)\; \times \; (0.20)$.
4. Step 4: Perform the multiplication to find the electric potential. Multiply the electric field strength by the radius of the sphere to determine the potential. This step involves simple arithmetic.
5. Step 5: Interpret the result. The calculated potential represents the electric potential on the surface of the sphere when electrical breakdown occurs. This is the maximum potential the sphere can sustain before ionization of the surrounding air occurs.

Key Concepts

Electric Potential

Electric potential, often referred to as voltage, is the amount of electric potential energy per unit charge at a point in an electric field. It indicates how much work would be done to move a charge from a reference point (usually at infinity) to that point. In the context of the Van de Graaff generator, the electric potential on the surface of the sphere is crucial for understanding how much energy is available to do work on charges in the vicinity.

Electric Field Strength

The electric field strength (E) is a measure of the force per unit charge experienced by a positive test charge placed in the field. It is defined as the gradient of the electric potential and is expressed in volts per meter (V/m). In the case of the Van de Graaff generator, the electric field strength at the surface of the sphere is critical for determining when electrical breakdown occurs, which is the point at which the air becomes ionized and conductive.

Breakdown Voltage

Breakdown voltage is the minimum voltage that causes a portion of an insulator to become electrically conductive. In the context of the Van de Graaff generator, it refers to the voltage at which the electric field strength exceeds the dielectric strength of air (approximately 3 x 10⁶ V/m), leading to ionization and the formation of a conductive path. This concept is essential for calculating the maximum electric potential on the surface of the sphere before breakdown occurs.