A Van de Graaff generator is a device for generating a large electric potential by building up charge on a hollow metal sphere. A typical classroom-demonstration model has a diameter of 30 cm. What is the electric field strength just outside the surface of the sphere when it is charged to 500,000 V?

Verified step by step guidance

1. Step 1: Understand the relationship between electric potential (V), electric field (E), and the radius (r) of the sphere. For a charged sphere, the electric field just outside the surface is given by the formula: $E=\frac{V}{r}$, where V is the electric potential and r is the radius of the sphere.
2. Step 2: Convert the diameter of the sphere to its radius. The diameter is given as 30 cm, so the radius is half of that: $r=\frac{30}{2}=\; 15$ cm. Convert this to meters since SI units are required: $r=\; 0.15$ m.
3. Step 3: Substitute the given values into the formula for the electric field. The electric potential is $V=\; 500,000$ V, and the radius is $r=\; 0.15$ m. The formula becomes: $E=\frac{\mathrm{500,000}}{0.15}$.
4. Step 4: Simplify the expression to calculate the electric field strength. Perform the division to find the value of $E$ in units of volts per meter (V/m).
5. Step 5: Verify the units and ensure the result is reasonable. The electric field strength should be expressed in V/m, and the magnitude should align with the expected range for a Van de Graaff generator.

Key Concepts

Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. It is defined as the force per unit charge and is measured in volts per meter (V/m). The strength of the electric field just outside a charged conductor can be calculated using the formula E = V/d, where V is the electric potential and d is the distance from the surface.

Electric Potential

Electric potential, measured in volts (V), 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 to a specific point in the field. In the context of the Van de Graaff generator, a potential of 500,000 V means that a charge would gain significant energy when moved to the surface of the sphere.

Conductors and Charge Distribution

Conductors allow electric charges to move freely, leading to a uniform distribution of charge on their surfaces when in electrostatic equilibrium. For a spherical conductor, the charge resides entirely on the surface, and the electric field outside the sphere can be treated as if all the charge were concentrated at the center. This principle simplifies calculations of electric fields around charged conductors.