Peoples Physics Concepts

http://www.ck12.org Chapter 14. Magnetism

Solution

In this problem, it is best to start by determining the direction of the force on each segment of the loop. Based on
the first right hand rule, the magnetic field from the infinite cable points into the page where the loop is. This means
that the force on the top segment of the loop will be down toward the bottom of the page, the force on the left
segment will be to right, the force on the bottom segment will be toward the top of the page, and the force on the
right segment will be to the left. The forces on the left and right segments will balance out because both segments
are the same distance from the cable. The forces from the top and bottom section will not balance out because the
wires are different distances from the cable. The force on the bottom segment will be stronger than the one on the
top segment because the magnetic field is stronger closer to the cable, so the net force on the loop will be up,
toward the top of the page.
Now we will begin to calculate the force’s magnitude by first determining the strength of the magnetic field at the
bottom and top segments. All we really have to do is plug in the distances to each segment into the equation we
already know for the magnetic field due to a current carrying wire.

B=

μoI

2 πr

Bbottom=

μoI 1

2 πR

Bto p=

μoI 1

2 π 2 R

Now we will calculate the net force on the loop using the equation given above. We’ll consider up the positive

direction.

ΣF=Fbottom−Fto p start by summing the forces on the loop

ΣF=I 2 LBbottom−I 2 LBto p substitute in the values for each of the force terms

ΣF=I 2 L(Bbottom−Bto p) factor the equation

ΣF=I 2 L(

μoI 1

2 πR

−

μoI 1

2 π 2 R

) substitute in the values for the magnetic field

ΣF=

μoI 1 I 2 L

2 πR

( 1 −

1

2

) factor the equation again

ΣF=

μoI 1 I 2 L

4 πR

simplify to get the answer

Watch this Explanation

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