A "ribbon" (width $a$) with uniform surface current density $K$ passes through a uniform magnetic field $\mathbf{B}_{ext}$. Only the length $b$ along the ribbon is in the field. What is the magnitude of the force on the ribbon?
<img src="./images/force_on_k_B.png" align="right" style="width: 500px";/>
1. $KB$
2. $aKB$
3. $abKB$
4. $bKB/a$
5. $KB/(ab)$
Note:
* CORRECT ANSWER: C
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<img src="./images/linecurrent_plain.png" align="right" style="width: 400px";/>
To find the magnetic field $\mathbf{B}$ at P due to a current-carrying wire we use the Biot-Savart law,
$$\mathbf{B}(\mathbf{r}) = \dfrac{\mu_0}{4\pi}I\int \dfrac{d\mathbf{l}\times\hat{\mathfrak{R}}}{\mathfrak{R}^2}$$
What is the direction of the infinitesimal contribution $\mathbf{B}(P)$ created by current in $d\mathbf{l}$?
1. Up the page
2. Directly away from $d\mathbf{l}$ (in the plane of the page)
3. Into the page
4. Out of the page
5. Some other direction
Note:
* CORRECT ANSWER: C
What is the magnitude of $\dfrac{d\mathbf{l}\times\hat{\mathfrak{R}}}{\mathfrak{R}^2}$?
<img src="./images/linecurrent_R_shown.png" align="right" style="width: 400px";/>
1. $\frac{dl \sin\theta}{\mathfrak{R}^2}$
2. $\frac{dl \sin\theta}{\mathfrak{R}^3}$
3. $\frac{dl \cos\theta}{\mathfrak{R}^2}$
4. $\frac{dl \cos\theta}{\mathfrak{R}^3}$
5. something else!
Note:
* CORRECT ANSWER: A
What is the value of $I \dfrac{d\mathbf{l} \times \hat{\mathfrak{R}}}{\mathfrak{R}^2}$?
<img src="./images/linecurrent_y0.png" align="right" style="width: 300px";/>
1. $\frac{I\,y\,dx'}{[(x')^2+y^2]^{3/2}}\hat{z}$
2. $\frac{I\,x'\,dx'}{[(x')^2+y^2]^{3/2}}\hat{y}$
3. $\frac{-I\,x'\,dx'}{[(x')^2+y^2]^{3/2}}\hat{y}$
4. $\frac{-I\,y\,dx'}{[(x')^2+y^2]^{3/2}}\hat{z}$
5. Other!
Note:
* CORRECT ANSWER: D
What do you expect for direction of $\mathbf{B}(P)$? How about direction of $d\mathbf{B}(P)$ generated JUST by the segment of current $d\mathbf{l}$ in red?
<img src="./images/curvy_wire_current.png" align="center" style="width: 400px";/>
1. $\mathbf{B}(P)$ in plane of page, ditto for $d\mathbf{B}(P$, by red$)$
2. $\mathbf{B}(P)$ into page, $d\mathbf{B}(P$, by red$)$ into page
3. $\mathbf{B}(P)$ into page, $d\mathbf{B}(P$, by red$)$ out of page
4. $\mathbf{B}(P)$ complicated, ditto for $d\mathbf{B}(P$, by red$)$
5. Something else!!
Note:
* CORRECT ANSWER: C
What is the magnitude of $\dfrac{d\mathbf{l}\times\hat{\mathfrak{R}}}{\mathfrak{R}^2}$?
<img src="./images/ringcurrent_R.png" align="right" style="width: 400px";/>
1. $\frac{dl \sin\phi}{z^2}$
2. $\frac{dl}{z^2}$
3. $\frac{dl \sin\phi}{z^2+a^2}$
4. $\frac{dl}{z^2+a^2}$
5. something else!
Note:
* CORRECT ANSWER: D
What is $d\mathbf{B}_z$ (the contribution to the vertical component of $\mathbf{B}$ from this $d\mathbf{l}$ segment?)
<img src="./images/ringcurrent_R.png" align="right" style="width: 400px";/>
1. $\frac{dl}{z^2+a^2}\frac{a}{\sqrt{z^2+a^2}}$
1. $\frac{dl}{z^2+a^2}$
1. $\frac{dl}{z^2+a^2}\frac{z}{\sqrt{z^2+a^2}}$
1. $\frac{dl \cos \phi}{\sqrt{z^2+a^2}}$
5. Something else!
Note:
* CORRECT ANSWER: A