Magnetic field components of a circuit

[Erik P]

Homework Statement

A circle-formed circuit with an area A lies in the (xy)-plane. An electric current I runs clockwise through the circuit (figure 1). An external magnetic field B exerts a torque t = D(3i - 4j) on the circuit, where D is a positive constant, i is the x-axis unit vector and j is the y-axis unit vector.

a.) Determine the circuits magnetic moment u and determine the magnetic field's components B_x and B_y.

Homework Equations

Right hand rule

\begin{equation}
\mu = IA
\end{equation}

\begin{equation}
\vec{\tau} = \vec{\mu} X \vec{B}
\end{equation}

The Attempt at a Solution

Solved the first part using the right hand rule, the magnetic moment is in the -z axis so we get:

\begin{equation}
\vec{\mu} = -IAk
\end{equation}

For the second part I know what it is supposed to be, but I get something different.

\begin{equation}
B_x = \frac{3}{IA}
\end{equation}

\begin{equation}
B_y = -\frac{4}{IA}
\end{equation}

But those answers are incorrect, the real answers have the 4 and 3 traded around and both are positive, instead of B_y being negative. Anyone able to tell me why that is? I figure that since 3 is multiplied by the x-axis unit vector that it should then be a part of B_x not B_y.

 

[Guneykan Ozgul]

Make sure that you are doing the cross product correctly. The other equations seems correct. It seems the mistake is in the cross product. When you cross i×k you should get -j and when you cross j×k you should get i.

 

[Erik P]

Make sure that you are doing the cross product correctly. The other equations seems correct. It seems the mistake is in the cross product. When you cross i×k you should get -j and when you cross j×k you should get i.

Redid the math, sure enough I made a mistake. Quick question though, should D not be included in the components? The answer says not, but I'm not sure on the reasoning.

 

[Guneykan Ozgul]

I guess you should include D's in the components otherwise how can you obtain a torque that contains D?

 

[Erik P]

I guess you should include D's in the components otherwise how can you obtain a torque that contains D?

Alright thanks, must be a typo in the set.

 

[Erik P]

I guess you should include D's in the components otherwise how can you obtain a torque that contains D?

Are you by any chance familiar with conversion of electrical energy to heat in coils/solenoids? Can't seem to figure out how to solve a problem involving that. Basically I need to calculate the total energy converted to heat in a solenoid for t>=, it's the last part of a 3 part problem, already solved the other 2 parts. I can't seem to find anything in my book or online about heat generation of coils. Could use an equation/law that describes it or something that I can convert into an expression that describes it.