How to find the magnetic field and magnetic force due to a solenoid loop

[Davidllerenav]

Homework Statement

For the rectangular coil of $N$ turns and with
the dimensions shown in figure 1,
calculate the magnetic field B that occurs
at a solid conductor point of length $l$
at a distance d from the coil. Consider that
The current flowing through the coil is $I_2$.

Relevant Equations

Biot-Savart Law: $\vec B= \frac{\mu_0}{2\pi}\int \frac{d\vec l \times \hat r}{r^2}$

I'm not so sure how to begin with this problem. I was thinking of usign superposition. I think that the field on the conductor due to the parallel segments of the coil is zero, since Ampere's Law tells us that the field outside the solenoid is zero, right? For the perpendicular segments, I used the field of a ring of radius $R$ at a point a distance $d$ from its center and I integrated, getting $\vec B= \frac{\mu_0nI_2}{2\pi}\int_0^l \frac{R^2dz}{(R^2+d^2)^{3/2}}=\frac{\mu_0nI_2}{2}\cdot \frac{l^2}{(l^2+R^2)^{1/2}}\hat z$. Am I correct?

 

[rude man]

Homework Statement: For the rectangular coil of $N$ turns and with
the dimensions shown in figure 1,
calculate the magnetic field B that occurs
at a solid conductor point of length $l$
at a distance d from the coil. Consider that
The current flowing through the coil is $I_2$.
Homework Equations: Biot-Savart Law: $\vec B= \frac{\mu_0}{2\pi}\int \frac{d\vec l \times \hat r}{r^2}$

View attachment 253023
I'm not so sure how to begin with this problem. I was thinking of usign superposition. I think that the field on the conductor due to the parallel segments of the coil is zero, since Ampere's Law tells us that the field outside the solenoid is zero, right? For the perpendicular segments, I used the field of a ring of radius $R$ at a point a distance $d$ from its center and I integrated, getting $\vec B= \frac{\mu_0nI_2}{2\pi}\int_0^l \frac{R^2dz}{(R^2+d^2)^{3/2}}=\frac{\mu_0nI_2}{2}\cdot \frac{l^2}{(l^2+R^2)^{1/2}}\hat z$. Am I correct?

Biot-Savart is the general law for finding the B field due to arbitrary current distributions.
You want the force exerted on a wire carring current due to an externally applied B field (like that of a second wire, hint hint). This is a vector relation so if for example the B field is generated by a second wire you need to consider the relative orientation between the two wires.
What you're looking for is in every intro physics text. Better yet, you can derive it yourself by starting with the expression for the Lorentz force F = qv x B and yes, combining with Ampere's law.. Forces can be added (vectorially of course).