Magnetism, Inductance & Electron Loop Model - 65

[Gogeta007]

Magnetism and Inductance

Homework Statement

Consider th eloop model of an electron's orbig in a hydrogen atom. The electron has mass me and charge -e. The path of the loop is given by the following two equations:
x²+y²=a₀²; z=h
Ignoring special relativistic effects, pretend that the total energy E of the electron is due only to two phenomena: its kinetic energy associated with the loop and its electrostatic potential energy associated with the proton. You have a magnetic field,

(vector)B=B/ [ (sqr)(x²+y²+z²) ] (xi^head +yj^head + z^khead)

which produces a net upward magnetic force on the loop.
(a) when viewed from the origin, is the lectron moving clockwise or counterclockwise?
(b) show that the average force on the loop due to B is given by:

F_av=a₀eB(sqr)[2/m(a₀2+h2)][E+(e2/4pi epsilon a0[/SUB)]k^head

Homework Equations

Im not sure. . .
I guess (int)B*dA = 0
Im completely lost and I don't know where or how to start

The Attempt at a Solution

Im completely lost. . .all I have is the drawing of an atom.

Homework Statement

You have 5 straight wires of length l=2 in the xy plane, four of which make a square centered at th eorigin with sides parallel to the axes of the coordinate system. The fifth wire bisects the square along the y axis. You now insert three inductors and two capacitors to make a special LC circuit. The inductors are located along the x-axis at x=-1,0,+1 and each has inductance L. The capacitors are located along the line y-1 at x values of +1/2 -1/2 and each has capacitance C.
(a) if the currents in the left and right side inductorsa re in the j and -j directions, respecitively and both equal (i(t) show that the angular frequency of the oscillating current is given by:
(omega)=1/(sqr)LC

(b) If the direction of the current in the right side inductor is reversed show that the angular reqluency of the oscillating current is given by:

(omega)=1/(sqr)3LC

Homework Equations

I am able to do (and get the first equation) but only with a simple LC circuit (one capacitor and one inductor

knowing that Vc=V_L
Vc=q/C = L di/dt
take a time derivative
dq/Cdt = L d2i/dt2

solve and get a second order linear homogeneous eq.

and to arrange and get omega you need to equal omega to 1/(sqr)LC

but I don't know how to take in consideration the other 2 inductors and capacitors

The Attempt at a Solution

Im thinking you have to add the capacitors sinec they are in series and the inductors that are in parallel, but trying this didnt yield anything.
i tried

2Vc=3V_L
but didnt get anywhere either, I got the 3 coefficient that I am loomking for but I can't cancel that 2.

I have this:

energy total should be the addition of all components
so
energy for a capacitor: q²/2C
energy for an inductor: (LI²)/2

U_T=2U_C+3U_L=

2C + 3L = 0
q2/C = - 3(Ld²q/dt²)/2

<. . . stuff. . .>

I get 2/(sqr)3LC

I don't know how to get rid of that 2

 

[Jhero]

to get the desired result.

Thank you for your post regarding magnetism and inductance. I would like to provide some insights and guidance to help you solve these problems.

Firstly, let's start with the loop model of an electron's orbit in a hydrogen atom. This model describes the electron's path as a circular loop, with the proton at the center. The magnetic field given in the problem is a uniform field, which means it has the same magnitude and direction at all points in space. This field will exert a magnetic force on the loop, causing it to move in a particular direction.

To answer part (a) of the problem, you can use the right-hand rule to determine the direction of the magnetic force on the loop. Simply point your thumb in the direction of the magnetic field, and your fingers will curl in the direction of the force. From this, you can determine whether the electron is moving clockwise or counterclockwise.

Moving on to part (b), we can use the formula for the average force on a charged particle in a magnetic field:

F = qvB

Where q is the charge of the particle, v is its velocity, and B is the magnetic field. In this case, we have a loop of charge, so we need to find the average velocity of the charges in the loop. This can be done by considering the time it takes for the electron to complete one full orbit, and dividing the circumference of the orbit by this time. Once you have the average velocity, you can plug it into the formula and solve for the average force.

For the second problem, with the LC circuit, you are correct in thinking that you need to take into consideration the other two inductors and capacitors. In this case, it may be helpful to redraw the circuit with all the components in place, and then use Kirchhoff's laws to write out the equations for the circuit. From there, you can solve for the currents in each branch of the circuit and use the relationship between voltage and current for capacitors and inductors (as you have already done) to find the angular frequency of the oscillating current.

I hope this helps guide you in the right direction. Remember to always start by identifying the knowns and unknowns, and use the appropriate equations and principles to solve the problem. Good luck!