An electron orbits a 3.0 cm diameter metal ball 5.0 mm above the surface. The orbital period of the electron is l2.0 micro seconds. What is the charge on the metal ball?
I think I use E=kq/r*R but im unsure!
Please help!
Thanks
To find the charge on the metal ball, you can use the equation for the centripetal force acting on the electron as it orbits the ball.
The centripetal force (Fc) exerted on the electron is given by the equation:
Fc = (me * v^2) / r
Where:
- me is the mass of the electron (9.11 x 10^-31 kg)
- v is the velocity of the electron
- r is the radius of the orbit (5.0 mm = 0.005 m)
We can find the velocity of the electron using the formula for the circumference of a circle:
C = 2 * π * r
The orbital period (T) is given as 2.0 microseconds (2.0 x 10^-6 s), which is the time taken to complete one full orbit.
So, T = C / v
v = C / T
Substituting the value of C from earlier:
v = (2 * π * r) / T
Now, substitute the value of v into the centripetal force equation:
Fc = (me * ((2 * π * r) / T)^2) / r
Simplifying gives:
Fc = (4 * π^2 * me * r) / T^2
The centripetal force is also equal to the electrical force (Fe) between the electron and the metal ball. This force is given by Coulomb's law:
Fe = (k * q1 * q2) / r^2
Where:
- q1 is the charge on the electron (−1.6 x 10^-19 C)
- q2 is the charge on the metal ball (which we want to find)
- r is the separation distance between the electron and the metal ball (3.0 cm = 0.03 m)
- k is the electrostatic constant (8.99 x 10^9 N*m^2/C^2)
Now, set the centripetal force equal to the electrical force:
(4 * π^2 * me * r) / T^2 = (k * q1 * q2) / r^2
Rearranging the equation to solve for q2 gives:
q2 = ((4 * π^2 * me * r) / T^2) * (r^2 / (k * q1))
Now, plug in the given values and calculate:
me = 9.11 x 10^-31 kg
r = 0.005 m
T = 2.0 x 10^-6 s
k = 8.99 x 10^9 N*m^2/C^2
q1 = -1.6 x 10^-19 C
Substitute these values into the equation and calculate q2 using a calculator.
q2 = ((4 * π^2 * (9.11 x 10^-31) * (0.005)) / (2.0 x 10^-6)^2) * (0.005^2 / (8.99 x 10^9 * (1.6 x 10^-19)))
Solve for q2 to get the charge on the metal ball.