- #1

2thumbsGuy

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## Homework Statement

A coil with N loops and radius R is surrounds a very long solenoid of radius r and n turns per meter.

The current in the solenoid is varying sinusoidally with time according to the relation I(t)=I

_{0}sin(2πft)

where I

_{0}is the maximum value of the current, and f is its frequency. (This is the type of current supplied by the power company.) If R >> r, show that the induced emf in the coil is given by

ε=-2π

^{2}μ

_{0}fNnr

^{2}I

_{0}cos(2πft)

## Homework Equations

I(t)=I

_{0}sin(2πft)

[itex]\Phi[/itex]B = ∫[itex]\vec{B}[/itex]∙d[itex]\vec{A}[/itex]

B = [itex]\mu[/itex]

_{0}nI

ε=N|(d[itex]\Phi[/itex]B)[itex]/[/itex]dt|

## The Attempt at a Solution

Substituting the above,

ε=N|(d∫[itex]\mu[/itex]

_{0}nI

_{0}sin(2πft)∙d[itex]\vec{A}[/itex])[itex]/[/itex]dt|

ε=N[itex]\mu[/itex]

_{0}nI

_{0}|(d∫sin(2πft)∙d[itex]\vec{A}[/itex])[itex]/[/itex]dt|

∫sin(2πft) = -cos(2πft)/2π

I believe (d∫sin(2πft)∙d[itex]\vec{A}[/itex])[itex]/[/itex]dt is a double integral, but if I evaluate I just end up with stuff in the denominator, which does not match with the answer.

I have no intuition for dot products and only a passable understanding of integrals, mathematically. I understand what they do, what they are for in layman's terms, but expressing it in mathematical terms is my perennial weakness. For the love of Congressman Weiner, help me to understand!