Question

A group of science and engineering students embarks on a quest to make an electrostatic projectile launcher. For their first trial, a horizontal, frictionless surface is positioned next to the 12-cm-diameter sphere of a Van de Graaff generator, and a small, 5.0 g plastic cube is placed on the surface with its center 2.0 cm from the edge of the sphere. The cube is given a positive charge, and then the Van de Graaff generator is turned on, charging the sphere to a potential of 200,000 V in a negligible amount of time. How much charge does the plastic cube need to achieve a final speed of a mere 3.0 m/s? Does this seem like a practical projectile launcher?

Answer 1

Electric charge on the plastic cube: $1.3\cdot 10^{-7}C$

Explanation:

The electric potential around a charged sphere (such as the Van der Graaf) generator is given by

$V(r)=\frac{kQ}{r}$

where

k is the Coulomb's constant

Q is the charge on the sphere

r is the distance from the centre of the sphere

Here we have:

V = 200,000 V on the surface of the sphere, so at r = 12.0 cm

We need to find the voltage V' at 2.0 cm from the edge of the sphere, so at

r' = 12.0 + 2.0 = 14.0 cm

Since the voltage is inversely proportional to r, we can use:

$Vr=V'r'\\V'=\frac{Vr}{r'}=\frac{(200,000)(12.0)}{14.0}=171,429 V$

This is the potential at the location of the plastic cube.

Now we can use the law of conservation of energy, which states that the initial electric potential energy of the cube is totally converted into kinetic energy when the plastic cube is at infinite distance from the generator. So we can write:

$qV' = \frac{1}{2}mv^2$

where:

q is the charge on the plastic cube

V' is the potential at the location of the cube

m = 5.0 g = 0.005 kg is the mass of the cube

v = 3.0 m/s is the final speed of the cube

Solving for q, we find the charge on the cube:

$q=\frac{mv^2}{2V'}=\frac{(0.005)(3.0)^2}{2(171,429)}=1.3\cdot 10^{-7}C$

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