# Determine a constant K

## Homework Statement

Hello out there...

I've kinda figured this out, but I'm not quite sure how tbh.

I got this problem:

The length a is constant, but b varies in time like this:

$$$b\left( t \right)=a\left( 1+{{\left( \frac{t}{\tau } \right)}^{2}}-2{{\left( \frac{t}{\tau } \right)}^{3}} \right),$$$
where $\tau$ is a timeconstant. Besides that I know that for t < 0 then b = a, and for t > $\tau$ then b = 2a.

The magnetic fields produced by the current in the conductors (I1 and I2) gives a magnetic flux through the rectangular loop of:

$${{\Phi }_{B}}=K\cdot b\left( t \right)$$

Determine the constant K.

## Homework Equations

$$B=\frac{{{\mu }_{0}}I}{2\pi r}$$

$$d{{\Phi }_{B}}=BdA=\frac{{{\mu }_{0}}I}{2\pi }L\,dr,$$
where L is b(t)

## The Attempt at a Solution

What I've done is as following:

$$${{\Phi }_{B}}=\int_{a}^{3a}{BdA}=\int_{a}^{3a}{\frac{{{\mu }_{0}}{{I}_{1}}}{2\pi }b\left( t \right)\,dr}+\int_{a}^{3a}{\frac{{{\mu }_{0}}{{I}_{2}}}{2\pi }b\left( t \right)\,dr}=-\frac{a{{\mu }_{0}}\left( {{I}_{1}}+{{I}_{2}} \right)\ln \left( 3 \right)\left( 2{{t}^{3}}-3\tau {{t}^{2}}-{{\tau }^{3}} \right)}{2\pi {{\tau }^{3}}}$$$

Putting this equal the magnetic flux I know, and then solving for K, I get:

$$K=\frac{{{\mu }_{0}}\left( {{I}_{1}}+{{I}_{2}} \right)\ln \left( 3 \right)}{2\pi },$$
which supposedly is the correct answer according to my book.

But what I don't understand is, that when I tried the limits a to 2a, which seems more obviously to me, I get ln(2) instead of ln(3). So I don't understand why the limits should be a to 3a instead - if I've even done it correctly in the first place.

So I thought one of you might knew :)

Regards.

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