A bead with a mass of 0.085 g and a charge of 20 nC is free to slide on a vertical rod. At the base of the rod is a fixed 20 nC charge.

In equilibrium, at what height above the fixed charge does the bead rest?

To find the height above the fixed charge at which the bead rests in equilibrium, we can use the concept of electric potential energy. The equilibrium position occurs when the electric potential energy of the bead is at a minimum.

Here are the steps to find the equilibrium height:

Step 1: Determine the electric potential energy (PE) of the bead at different heights using the formula:

PE = k(|q1||q2|) / r

where k is the electrostatic constant (8.99 x 10^9 Nm²/C²), |q1| and |q2| are the absolute values of the charges, and r is the distance between the charges.

In this case, |q1| = |q2| = 20 nC = 20 x 10^-9 C.

Step 2: Calculate the electric potential energy of the bead at different heights above the fixed charge.

The electric potential energy of the bead can be calculated using the equation:

PE = mgh

where m is the mass of the bead, g is the acceleration due to gravity (9.8 m/s²), and h is the height above the fixed charge.

Step 3: Set up an equation to find the equilibrium height.

At equilibrium, the electric potential energy is at a minimum. Therefore, we need to find the height at which the potential energy is minimized. This occurs when the gravitational potential energy is equal to the electric potential energy:

mgh = k(|q1||q2|) / r

Substitute the known values:

0.085g x 9.8m/s² x h = (8.99 x 10^9 Nm²/C²) x (20 x 10^-9 C)^2 / r

Simplify the equation by canceling out units:

0.085 x 9.8 x h = 8.99 x (20)^2 / r

Step 4: Solve for the height.

To find the equilibrium height, rearrange the equation to solve for h:

h = (8.99 x (20)^2 / r) / (0.085 x 9.8)

Finally, substitute the value of r (which is the distance between the fixed charge and the bead) to find the height at which the bead rests in equilibrium above the fixed charge.

Note: The value of r is not given in the question, so you need to determine the distance between the fixed charge and the bead.

To find the height above the fixed charge at which the bead rests in equilibrium, we can use the principle of electrostatic equilibrium.

In electrostatic equilibrium, the net force acting on an object must be zero. Since the bead is free to slide on the vertical rod, the only forces acting on it are its weight and the electrostatic force between the two charges.

Let's break down the solution into steps:

Step 1: Calculate the weight of the bead.
The weight of an object can be found using the equation:
Weight = mass * acceleration due to gravity
Weight = 0.085 g * (9.8 m/s^2) (1 g = 9.8 m/s^2)
Weight = 0.085 g * 9.8 m/s^2
Weight = 0.833 N

Step 2: Calculate the electrostatic force between the two charges.
The electrostatic force between two charges can be found using Coulomb's Law:
Force = (k * |q1 * q2|) / r^2
where:
- k is the electrostatic constant (k = 8.99 x 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges
- r is the distance between the charges.

Force = (8.99 x 10^9 N m^2/C^2 * |20 nC * 20 nC|) / r^2
Force = (8.99 x 10^9 N m^2/C^2 * 20 nC * 20 nC) / r^2

Step 3: Set up equilibrium condition.
For the bead to be in equilibrium, the net force acting on it must be zero.
Net force = Electrostatic force - Weight
Setting Net force to zero:
(8.99 x 10^9 N m^2/C^2 * 20 nC * 20 nC) / r^2 = 0.833 N

Step 4: Solve for the height.
Now, we can solve the above equation to find the value of height (r). Rearranging the equation, we have:
r^2 = (8.99 x 10^9 N m^2/C^2 * 20 nC * 20 nC) / 0.833 N
r^2 = (8.99 x 10^9 N m^2/C^2 * 20 x 10^-9 C * 20 x 10^-9 C) / 0.833 N
r^2 = 8.58 x 10^-7 m^2

Taking the square root of both sides, we get:
r = 0.000928 m

Therefore, the bead rests at a height of approximately 0.000928 meters (or 0.928 millimeters) above the fixed charge.

Let the charge separation be x. The elecrostatic repulsion force must equal the weight.

k*(20*10^-9)^2/x^2 = M g

M = 85*10^-6 kg
g = 9.8 m/s^2
k is the Coulomb constant. (You may need to look it up. If you are studying elctricity, your text will say what the value is.)

Solve for x

Thank you so much!!!! you're a lifesaver!