Electrostatics, finding velocity of electron

[physics604]

1. An electron orbits the nucleus of an atom with velocity v. If this electron were to orbit the same nucleus with twice the previous orbital radius, its orbital velocity would now be

a) $\frac{v}{2}$
b) v
c) 2v
d) $\frac{v}{√2}$

Homework Equations

$\Delta$E_k + $\Delta$E_p = 0 ?

The Attempt at a Solution

I'm not really sure how I should start this question. None of the equations I have include both the v and the r variable.

 

[berkeman]

What is the equation for electrostatic attraction?

 

[physics604]

F = kQ₁Q₂ / r²

But I don't see how that helps.

 

[berkeman]

F = kQ₁Q₂ / r²

But I don't see how that helps.

And the equation that we use for centripital force for uniform circular motion? There is a reason that we ask you to list the Relevant Equations...

 

[Nickie]

Can you provide a hint or some guidance on how to approach this?

Sure, no problem. This problem involves the concept of conservation of energy in electrostatics. The equation you have written is a good starting point, as it represents the conservation of total energy in a system. In this case, the system is the electron orbiting the nucleus.

To solve this problem, you will need to use the equation for the electrostatic potential energy of a system:

U = \frac{kQq}{r}

where U is the potential energy, k is the Coulomb constant, Q and q are the charges of the two objects, and r is the distance between them.

You can also use the equation for the kinetic energy of a moving object:

K = \frac{1}{2}mv^2

where K is the kinetic energy, m is the mass of the object, and v is its velocity.

Now, you can set up an equation where the change in kinetic energy (\DeltaK) is equal to the negative change in potential energy (\DeltaU):

\DeltaK = -\DeltaU

You can also substitute the equations for kinetic and potential energy to get:

\frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2 = -\frac{kQq}{r_f} + \frac{kQq}{r_i}

where v_f is the final velocity, v_i is the initial velocity, r_f is the final orbital radius, and r_i is the initial orbital radius.

Now, you can use this equation to solve for the final velocity, v_f, in terms of the initial velocity, v_i. You can also use the given information that the final orbital radius is twice the initial orbital radius (r_f = 2r_i) to simplify the equation further.

I hope this helps guide you in solving this problem. Remember to always start with the relevant equations and use the given information to simplify the problem. Good luck!