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

To find the magnetic field $\mathbf{B}$ at P due to a current-carrying wire we use the Biot-Savart law, $$\mathbf{B}(\mathbf{r}) = \dfrac{\mu_0}{4\pi}I\int \dfrac{d\mathbf{l}\times\hat{\mathfrak{R}}}{\mathfrak{R}^2}$$ In the figure, with $d\mathbf{l}$ shown, which purple vector best represents $\mathfrak{R}$? <img src="./images/linecurrent_r.png" align="center" style="width: 400px";/> Note: * CORRECT ANSWER: A

<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