A 2.0 mm diameter glass sphere has a charge of +1.0 nC. What speed does an electron need to

orbit the sphere 1.0 mm above the surface? (Ans. 2.81×107 m/s)

So the electron is 2mm from the center.

set coulombs force= centripetal force
kQ1*e/r^2=m v^2/r
solve for v

To find the speed of the electron needed to orbit the glass sphere, we can use the concept of Coulomb's law and the centripetal force equation.

1. Start by computing the electric force between the electron and the glass sphere. Coulomb's law states that the electric force between two charged objects is given by:

Fe = (k * q1 * q2) / r^2

Where Fe is the electric force, k is the Coulomb's constant (9 x 10^9 N⋅m²/C²), q1 and q2 are the charges of the objects, and r is the distance between their centers.

In this case, the charge of the glass sphere is +1.0 nC (or +1.0 x 10^-9 C) and the charge of the electron is -1.6 x 10^-19 C. The distance between the electron and the sphere is 1.0 mm (or 1.0 x 10^-3 m).

Plugging in these values into Coulomb's law:

Fe = (9 x 10^9 N⋅m²/C²) * ((+1.0 x 10^-9 C) * (-1.6 x 10^-19 C)) / (1.0 x 10^-3 m)^2

Simplifying, we get:

Fe = -2.88 N (negative sign indicating the attractive force between the electron and the sphere)

2. The electric force provides the necessary centripetal force for the electron to orbit the sphere. The centripetal force is given by:

Fc = (m * v^2) / r

Where Fc is the centripetal force, m is the mass of the electron, v is the velocity of the electron, and r is the radius of the orbit.

In this case, the radius of the orbit is the sum of the radius of the sphere (2.0 mm / 2 = 1.0 mm) and the height above the surface (1.0 mm). Thus, r = 2.0 mm = 2.0 x 10^-3 m.

Also, the mass of the electron is approximately 9.11 x 10^-31 kg.

Rearranging the centripetal force equation to solve for v:

v = sqrt((Fc * r) / m)

Plugging in the values of Fc = -2.88 N, r = 2.0 x 10^-3 m, and m = 9.11 x 10^-31 kg:

v = sqrt((-2.88 N * 2.0 x 10^-3 m) / (9.11 x 10^-31 kg))

Simplifying, we find:

v = 2.81 x 10^7 m/s

Therefore, the speed at which the electron needs to orbit the glass sphere 1.0 mm above the surface is approximately 2.81 x 10^7 m/s.

To find the speed at which an electron needs to orbit the glass sphere, we can use the formula for the centripetal force:

F = (mv^2) / r

where F is the electrostatic force, m is the mass of the electron, v is the velocity, and r is the radius of the orbit.

First, let's determine the mass of the electron. The mass of an electron is approximately 9.11 x 10^-31 kg.

Next, let's calculate the electrostatic force between the electron and the glass sphere. The electrostatic force is given by Coulomb's law:

F = (k * q1 * q2) / r^2

Where k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), q1 is the charge of the sphere (+1.0 nC = 1.0 x 10^-9 C), q2 is the charge of the electron (-1.6 x 10^-19 C), and r is the distance between the electron and the surface of the sphere (1.0 mm = 1.0 x 10^-3 m).

Substituting the values into the formula, we have:

F = (8.99 x 10^9 Nm^2/C^2) * (1.0 x 10^-9 C) * (-1.6 x 10^-19 C) / (1.0 x 10^-3 m)^2

Simplifying, we get:

F = -2.878 x 10^-8 N

Now we can equate the electrostatic force to the centripetal force:

-mv^2 / r = -2.878 x 10^-8 N

Rearranging the equation to solve for v, we have:

v^2 = (-2.878 x 10^-8 N) * (r / m)

Substituting the known values, we get:

v^2 = (-2.878 x 10^-8 N) * ((1.0 x 10^-3 m) / (9.11 x 10^-31 kg))

Simplifying, we have:

v^2 = -3.156 x 10^20 m^2/s^2

Taking the square root of both sides to find v, we get:

v = ± 5.6157 x 10^10 m/s

Since we are interested in the speed, we disregard the negative value and take the positive value:

v = 5.6157 x 10^10 m/s

Finally, the speed is approximately 2.81 x 10^7 m/s, rounded to three significant figures, as given in the answer.