Lorentz force on current-carrying wire

[quasarpulse]

Homework Statement

Given: a quarter-circle of wire (radius r=0.75m) in a uniform 1.7T magnetic field carrying a current of 3.5A (see diagram)

Find: The force on the wire.

Note: the connecting wires delivering the current to the quarter circle are parallel to the magnetic field and experience no force.

Homework Equations

dF = I dS x B

The Attempt at a Solution

I know of two ways to solve this.

The first is to use the fact that the force is path-independent and use the F = I L x B formulation; we're not supposed to do that, as this is supposed to be a calculus problem.

The second is the way my instructor suggested to the class, which is to argue by symmetry that the force is directed at a 45 degree angle outward from the origin. I can then treat it like a scalar integration problem and use the fact that the magnitude of dS is r d(theta). It's really a quite simple problem this way, but it only works because the problem is particularly simple.

Having done vector calculus, I feel like there ought to be a third way to attack the thing; it should be possible to parameterize the curve and do something like a line integral. The advantage would be that the same approach would work for a messier problem, where e.g. the magnetic field was nonuniform so the direction wasn't plainly obvious. But I'm not quite sure where to begin. I've got the following parameterization:
x(t) = r sin t
y(t) = r cos t
z(t)=0
0 <= t <= pi/2
but I'm not sure where to go from there. Any ideas?

 

[Redbelly98]

I haven't thought this through completely, but it seems the arc-length formula would be useful:

dS = (dx² + dy²)^1/2 = (1 + (dy/dx)²)^1/2 dx
​

And then express dy and dx in terms of t and dt.

 

[nlightner]

I would first commend the student for their thorough understanding of the problem and for considering multiple approaches to solving it. I would also agree that there may be other ways to solve the problem using vector calculus, but the two methods mentioned are both valid and efficient ways to solve the problem.

If the student wants to try using a line integral approach, they could start by defining the position vector of a point on the wire as r(t) = (r cos t)i + (r sin t)j + 0k, where t is the parameter along the curve (in this case, the angle theta). They could then use the definition of the Lorentz force, F = I dS x B, and take the cross product of dS and B to get a vector expression for the force. Then, they could integrate this force vector along the curve using the limits of t = 0 to t = pi/2. This would result in the same answer as the previous methods, but it may provide a more generalizable approach for more complex problems.

Overall, it is important for a scientist to approach problems in multiple ways and to consider the most efficient and accurate method for each specific situation. In this case, both methods presented by the student are valid and effective, and the decision to use one over the other may depend on the specific problem at hand.