Question

use amperes law to solve for the magnetic field magnitude ( b ) along side #1 of the loop. this will be a symbolic expression in terms of any or all of the following: ( mu_0, n, i, l, w ), and ( r ).

( b=square )

Explanation:

Step1: State Ampere's law

$$\oint \vec{B}\cdot d\vec{l}=\mu_0I_{enc}$$

Step2: Analyze the loop for side #1

Assume the loop is chosen such that the magnetic field \(B\) is constant along side #1 and perpendicular to the other sides (so their dot - product with \(d\vec{l}\) is zero). Let the length of side #1 be \(l\). The current enclosed \(I_{enc}=n\cdot i\cdot l\) (where \(n\) is the number of turns per unit length). Then \(\oint \vec{B}\cdot d\vec{l}=B\cdot l\) (since \(B\) and \(d\vec{l}\) are parallel along side #1 and zero for other sides).

Step3: Solve for \(B\)

From Ampere's law \(B\cdot l=\mu_0\cdot n\cdot i\cdot l\). Divide both sides by \(l\) (assuming \(l
eq0\)).

Answer:

\(B = \mu_0ni\)