Help on Electricity and magnetism HW

[dsb199094]

Homework Statement

A force eld E has the spherical components
E_r = (2Dcosθ)/r3 E_θ = (Dsinθ)/r3 E$\phi$ = 0:

(a) Evaluate by line integration the work it does in taking a point (parti-
cle) from point A in the diagram to point B via the quarter circle r = a; 0 < θ < =2. (See accompanying diagram.)

(b) Do the same for the path that includes the sector r = b < a; 0 < θ < 3$\pi$/2.

(c) You should conclude that E is derivable from a scalar potentia via
E = -$\Phi$. Find in a gauge such that $\Phi$(r→∞) = 0.

(d) Show that each piece of work you evaluated in (a) and (b) equals the
potential dierence between the endpoints of the piece.

Homework Equations

a) w=$\int$E(dot product)dl

b) i can figure out once i get a

c)not sure how to get E from just having E=-∇0 or am i over thining this

d) need some sort of elaboration on this part not sure what it's asking.

The Attempt at a Solution

MY understanding of what I am doing would be a line integral so for parts a and b my guess would be to take the integral ∫_a^aEr_rdr + ∫₀^$\pi$/2E_θrdθ + ∫E_$\Phi$d$\Phi$

so it would come out to
∫₀^$\pi$/2E_θrdθ which simplifies to D/r²
also i can't find the equation used for the below question anywhere in my book just help with the equation would be great

Homework Statement

A neutral particle of mass m and a permanent dipole D (like a water
molecule) is placed at the center of a uniformly charged ring (radius a, charge
q > 0).

a) What is the minimum initial velocity u_m that the particle must be
given if it is to escape to infinity, tail first along the axis of the ring?

b) What happens if the particle is disturbed with less than that minimum
velocity?

 

[rude man]

It might seem like it's insufficient to just give the dipole moment, which is the product of one of the charges times distance of separation: D = qd. But, maybe if you do the math that turns out not to be the case.

So, treat the problem as two separate charges separated by d. The direction of travel towards infinity is such that the +q charge leads the -q charge so the net force on the dipole is always towards the charged ring, and positive force needs to be done to bring the dipole to infinity.

Note that the force on D is initially zero (at the location of the center of the charged disc.

Good problem!

Hint: since the E field obviously can't be uniform, would a gradient be involved in some way?

 

[rude man]

Homework Statement

A force eld E has the spherical components
E_r = (2Dcosθ)/r3 E_θ = (Dsinθ)/r3 E$\phi$ = 0:

(a) Evaluate by line integration the work it does in taking a point (parti-
cle) from point A in the diagram to point B via the quarter circle r = a; 0 < θ < =2. (See accompanying diagram.)D/r²

Find out if 0 < θ < π/2 isn't meant. 0 < θ < 2 does not correspond to the figure, going from A to B along radius a.

 

[dsb199094]

It might seem like it's insufficient to just give the dipole moment, which is the product of one of the charges times distance of separation: D = qd. But, maybe if you do the math that turns out not to be the case.

So, treat the problem as two separate charges separated by d. The direction of travel towards infinity is such that the +q charge leads the -q charge so the net force on the dipole is always towards the charged ring, and positive force needs to be done to bring the dipole to infinity.

Note that the force on D is initially zero (at the location of the center of the charged disc.

Good problem!

Hint: since the E field obviously can't be uniform, would a gradient be involved in some way?

the formula i was given today by my instructor was F = (p (dot product) ∇)E

He also said that I needed to find the potential first... now how do you take the reverse divergence as shown above for the p(dot)∇ and how would i find the potential of this by solving for E and pluging that E into V=-∫^b_a E dl??

 

[dsb199094]

Find out if 0 < θ < π/2 isn't meant. 0 < θ < 2 does not correspond to the figure, going from A to B along radius a.

sorry its supposed to be 0 < θ < π/2 i must have made a mistake typing it

 

[rude man]

the formula i was given today by my instructor was F = (p (dot product) ∇)E

He also said that I needed to find the potential first... now how do you take the reverse divergence as shown above for the p(dot)∇ and how would i find the potential of this by solving for E and pluging that E into V=-∫^b_a E dl??

The formula he gave is what I had in mind: F_x = D * grad E_x.

I don't see why you need to get the potential. In fact, I don't see what one would do with it.

You have an expression for E_x along the axis of the charged ring. Just go with that.

PS oh I suppose you could calculate the E field along the axis by calculating the potential first and then taking E = - grad V. But chances are your textbook already has an expression for E_x(x).

 

[dsb199094]

that makes since... so what would i do at this point... F_x = D * grad E_x and plug 1/2 m v² for E_x then solve for V

 

[rude man]

No, no. You can't equate energy with the electric field. Different units!

You compute F_x(x) = D*grad E_x
= D_x{grad E_x}_x since we're staying on the x axis.

Once you know F_x(x) you can integrate it and equate it to the initial kinetic energy.

Same idea as escape velocity from the surface of the Earth. How did you compute that?

 

[dsb199094]

sorry spent all day studying for another exam and guess i started to mix up variable ha...
so basically i sould get D_x(∇E_x)= F_x is D a constant at this point?

then by using the eqs
W=∫F and W=$\Delta$KE we get $\Delta$KE = ∫F = initial KE

then using what i made the mistake of doing last night finding V from the KE equation

so i should get somthing like this (I'm assuming D is constant)

(D∫∇E)=KE

but the escape velocity is when the KE and the gravitational potential energy is 0 and the formula is 1/2mv² + -Gm/r=0 so V_e= √(2Gm/r)

or would the escape velocity in this case be KE and just electric potential energy is 0
so
1/2mv²+-1/(4$\pi$ε₀)*(Qq/r)= 0 so then
V_e=√(-2/(4$\pi$ε₀)*(Qq/r)/m)

 

[rude man]

sorry spent all day studying for another exam and guess i started to mix up variable ha...
so basically i sould get D_x(∇E_x)= F_x is D a constant at this point?

It's ∫Fdx, not ∫F.

Yes, D is a constant, it's given and it has an x component only.

So your force is along the x-axis and is F = D grad E. grad E is a vector pointing along the x-axis and so has only an x component also.

Your next job is to get E which is also along the x-axis only. Compute it any way you want or look it up.

then by using the eqs
W=∫F and W=$\Delta$KE we get $\Delta$KE = ∫F = initial KE

then using what i made the mistake of doing last night finding V from the KE equation

so i should get somthing like this (I'm assuming D is constant)

(D∫∇E)=KE

Yes, if you fix your integral.

but the escape velocity is when the KE and the gravitational potential energy is 0 and the formula is 1/2mv² + -Gm/r=0 so V_e= √(2Gm/r)

or would the escape velocity in this case be KE and just electric potential energy is 0
so
1/2mv²+-1/(4$\pi$ε₀)*(Qq/r)= 0 so then
V_e=√(-2/(4$\pi$ε₀)*(Qq/r)/m)

I did not intend for you to invoke any gravitational formulas. I wanted you to think of how the escape velocity is derived without using potentials. It's derived by integrating the gravitational force from the radius of the Earth to infinity. Here your force is not gravitational but electrostatic, and you will be integrating from x = 0 to x = ∞. Don't try to mix potentials into the computations. It's beside the point.

In both cases the integral of force over the relevant distance is equated to the initial k.e. as you say.

 

[dsb199094]

So I will take this ( D∫∇E)=KE and solve for v by putting in 1/2mv^2 for KE how did you go from (D (dot) gradient)E to D(gradient E)

 

[rude man]

So I will take this ( D∫∇E)=KE and solve for v by putting in 1/2mv^2 for KE how did you go from (D (dot) gradient)E to D(gradient E)

The vector equation for the x component of force is F_x = D*grad E_x. But for you, D = D i so the vector equation reduces to D times the magnitude of grad E_x.

D is just a constant vector. D is its magnitude. grad E_x is also a vector. This vector is a function of x. But both vectors point in the x (in the i) direction so the dot-product is simply the product of their magnitudes.

 

[dsb199094]

ok thanks for all the help