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Homework Statement
A force eld E has the spherical components
E_{r} = (2Dcosθ)/r3 E_{θ} = (Dsinθ)/r3 E[itex]\phi[/itex] = 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[itex]\pi[/itex]/2.
(c) You should conclude that E is derivable from a scalar potentia via
E = [itex]\Phi[/itex]. Find in a gauge such that [itex]\Phi[/itex](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=[itex]\int[/itex]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 im doing would be a line integral so for parts a and b my guess would be to take the integral ∫_{a}^{a}Er_{r}dr + ∫_{0}^{[itex]\pi[/itex]/2}E_{θ}rdθ + ∫E_{[itex]\Phi[/itex]}d[itex]\Phi[/itex]
so it would come out to
∫_{0}^{[itex]\pi[/itex]/2}E_{θ}rdθ which simplifies to D/r^{2}
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?
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