Question

A 2.6 mm diameter sphere is charged to -4.4 nC. An electron fired directly at the sphere from far away comes to within 0.31 mm of the surface of the target before being reflected. What was the electron's initial speed?

Answer 1

The electron's initial speed is approximately 2.76 x 10^7 m/s.

Step-by-step explanation:

To solve this problem, we can use the principles of electrostatic potential energy and kinetic energy. The electrostatic potential energy gained by the electron as it approaches the sphere is equal to the kinetic energy it possesses just before being reflected.

The electrostatic potential energy (U) can be calculated using the formula:

U = {k . q_1 . q_2} / {r}

where:

- ( k ) is Coulomb's constant (8.99 x 10^9 N m^2/ C^2),

- ( q_1 ) and ( q_2 ) are the magnitudes of the charges,

- ( r ) is the separation between the charges.

The kinetic energy (K) can be calculated using the formula:

K = 1 / 2 m v^2

where:

- m is the mass of the electron (9.11 x 10^-31 kg),

- v is the speed of the electron.

Setting (U) equal to (K), we get:

{k . q . q_e / r = 1 / 2 m v^2

Solving for (v):

v = (2 . k . q . q_e / m . r)^1/2

Substitute the known values into the equation to find \(v\). Note that the charge of an electron is q_e = -1.6 x 10^-19 C.

v = {{2 . (8.99 x 10^9) . (-4.4 x 10^{-9}) .(1.6 x 10^{-19})}^1/2 / {(9.11 x 10^{-31}) . (0.31 x 10^{-3})}}

After calculating this expression, you'll find the initial speed v of the electron.

Answer 2

To find the electron's initial speed, we will use the principle of conservation of energy.

Step-by-step explanation:

To find the electron's initial speed, we will use the principle of conservation of energy. We can equate the initial kinetic energy of the electron to the potential energy when it is at its closest distance to the charged sphere.

1. Calculate the potential energy using the formula PE = k*q1*q2/r, where k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2), q1 is the charge of the sphere (-4.4 x 10^-9 C), q2 is the charge of the electron (-1.6 x 10^-19 C), and r is the distance between them (0.31 mm).

2. Equate the potential energy to the initial kinetic energy of the electron, which is given by KE = 1/2*m*v^2, where m is the mass of the electron (9.11 x 10^-31 kg) and v is its initial speed.

3. Solve for the initial speed (v).