What Speed Must an Electron Achieve to Orbit a Charged Glass Sphere?

[tomizzo]

Homework Statement

A 2.70-mm-diameter glass sphere has a charge of + 1.10 nC.

What speed does an electron need to orbit the sphere 1.50mm above the surface?

Homework Equations

a = v^2/r -> force = m*v^2/r

electrostatic force = K*Q1*Q2/distance^2

therefore:

m*v^2/r=K*Q1*Q2/distance^2

The Attempt at a Solution

(9.10938291 × 10-31 kg)(v^2)/((2.7*10^-3)/2) = 8.99*10^9*(1.60*10^-19)*(1.1*10^-9)/(1.50*10^3)^2

I get 32282518 m/s which is the incorrect answer. However, I just noticed something. The radius I'm using is half of the sphere, should I be adding the separation distance to the radius as well?

 

[lightgrav]

yes, the circle's radius is bigger than the glass radius.

 

[tomizzo]

yes, the circle's radius is bigger than the glass radius.

I got an answer of 4.69*10^7 m/s and it still says it's incorrect.

 

[lightgrav]

I didn't get that speed. why don't you cancel one of the radius variables, and try the calculation again.

 

[tomizzo]

Here are my values:

Mass of electron = 9.10938*10^-31 kg
v = ?
radius = half of glass sphere plus separation distance = (2.7/2)*10^-3+1.5*10^-3
K = 8.99*10^9
Q1 = 1.6*10^-19 (charge of electron in coulombs)
Q2 = 1.1*10^-9
distance = 1.5*10^-3

Now I'm realizing the distance should probably be (2.7/2)*10^-3+1.5*10^-3 also...

 

[tomizzo]

With the change, I'm getting an answer of 2.469*10^7 m/s. Does this look familiar?

 

[lightgrav]

I think it rounds up to 24.7 Mm/s ... the E-field has spread more at that distance, so is weaker at the electron.

 

[Herman_in_the_North]

Well, this is obviously way too late, but for all of you physics nerds out there here it is.

Uniformly Charged Sphere Equation:

E=Q/(e0*4*r^2*pi)

E=electric field
Q=charge of sphere 1.1*10^-9 (in this case)
e0=8.85*10^-12
r=1.5*10^-3 (in this case)
pi = pi ;)

Take that E and plug it into the following Electric Force Equation:

E=F/q

E= what you solved for previously
F=Force that you want to derive
q=1.6*10^-19 (constant for charge of electron)

Take that F and plug it into this standard Force Equation:

F=ma

F=what you solved for previously
m=9.11*10^-31
a=what you want to solve forLast, but not least plug the a you solved for into the following Uniform Circular Motion Equation:

a=v^2/r

a=what you just solved for
v=what you want to solve for
r=1.5*10^-3 (in this case)That's it ladies and gents. Physics = MAGIC