Calculating Charge & Radius of a Van der Graaff Generator Dome

[riincanavi1]

1. A spherical dome of a Van der Graaff genenerator can be raised to a maximum electrical
potential of 600 kV; then additional charge leaks off in sparks as the electric field reaches 3.0
MV/m. Determine the charge and radius of the dome.



2. Homework Equations [E=kq/r^2 and V=kq/r]



3. I don't know how to relate the given Voltage and the given E field to find charge and radius

 

[Redbelly98]

You'll use the equations you quoted, for the values given in the problem statement.

I.e.,

3.0 MV/m = kq/r^2
600 kV = kq/r

Work with those to get r and q.

 

[tomcue]

.

I can provide a response by explaining the relationship between voltage, electric field, charge, and radius in a Van der Graaff generator.

Firstly, the Van der Graaff generator works by using a moving belt to transfer charge from a high-voltage power source to a spherical dome. As the charge accumulates on the dome, the electric field increases until it reaches a critical point where the charge starts to leak off in sparks. This critical point is known as the breakdown voltage and it is typically around 3.0 MV/m for a Van der Graaff generator.

Using the given information, we can use the equations for electric field and voltage to determine the charge and radius of the dome. The electric field equation (E=kq/r^2) relates the electric field (E) to the charge (q) and the radius (r) of the dome. The voltage equation (V=kq/r) relates the voltage (V) to the charge (q) and the radius (r).

To find the charge, we can rearrange the voltage equation to solve for q: q=Vr/k. Substituting the given voltage of 600 kV and using the value of k, which is the Coulomb's constant (k=8.99x10^9 Nm^2/C^2), we can calculate the charge to be 6.68x10^-7 C.

To find the radius, we can rearrange the electric field equation to solve for r: r=sqrt(kq/E). Substituting the calculated charge and the given electric field of 3.0 MV/m, we can calculate the radius to be 0.015 m or 1.5 cm.

In conclusion, the calculated charge and radius of the Van der Graaff generator dome are 6.68x10^-7 C and 1.5 cm, respectively. It is important to note that these calculations are based on ideal conditions and may vary in real-life situations. Additionally, factors such as humidity, temperature, and the material of the dome can also affect the values.