An infinitely long straight conducting cylinder of radius 4R lies along the z-axis. The current density in the cylinder is J along the -z axis and uniform over the cross-section. Find the magnitude and direction of the magnetic field at point A. Select one: a.??JR, +x-axis b.??JR, -x-axis c.2??JR, +y-axis d.??JR, +y-axis e.2??JR, -x-axis f.??JR, -y-axis g.2??JR, -y-axis h.2??JR, +x-axis
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The current density J is the current per unit area, so the total current I is given by I = J * Area. The cross-sectional area of the cylinder is π*(4R)^2 = 16πR^2, so I = J * 16πR^2. Show more…
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A long, straight solid cylinder with radius R, oriented with its axis in the z direction, carries a total current I0 whose current density is J. The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relationship: J = c[1 - (r/R)^2]k for r <= R J = 0 for r > R (a) Determine the constant c in terms of I0 and R, or a subset of these quantities (your answer may have numerical factors). (b) What are the magnitude and direction (clockwise or counterclockwise) of the magnetic field at a distance r > R? (c) What are the magnitude and direction of the magnetic field at a distance r < R?
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A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is $J .$ The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relationship $$ \begin{array}{rlrl}{\overrightarrow{\boldsymbol{J}}} & {=\frac{2 I_{0}}{\pi a^{2}}\left[1-\left(\frac{r}{a}\right)^{2}\right] \hat{\boldsymbol{k}}} & {} & {\text { for } \boldsymbol{r} \leq \boldsymbol{a}} \\ {} & {=\mathbf{0}} & {} & {\text { for } \boldsymbol{r} \geq a}\end{array} $$ where $a$ is the radius of the cylinder, $r$ is the radial distance from the cylinder axis, and $I_{0}$ is a constant having units of amperes. (a) Show that $I_{0}$ is the total current passing through the entire cross section of the wire. (b) Using Ampere's law, derive an expression for the magnitude of the magnetic field $\vec{B}$ in the region $r \geq a$ . (c) Obtain an expression for the current $I$ contained in a circular cross section of radius $r \leq a$ and centered at the cylinder axis. (d) Using Ampere's law, derive an expression for the magnitude of the magnetic field $\vec{B}$ in the region $r \leq a$ . How do your results in parts $(b)$ and $(d)$ compare for $r=a ?$
A long, straight, solid cylinder, oriented with its axis in the $z$-direction, carries a current whose current density is $\overrightarrow{J}$. The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship $$\overrightarrow{J} = \frac{2I_0}{\pi{a}^2} [1-(\frac{r}{a})^2]\hat{k} \space for \space r \leq a$$ $$=0 \space for \space r \geq a$$ where a is the radius of the cylinder, $r$ is the radial distance from the cylinder axis, and $I_0$ is a constant having units of amperes. (a) Show that $I_0$ is the total current passing through the entire cross section of the wire. (b) Using Ampere's law, derive an expression for the magnitude of the magnetic field $\overrightarrow{B}$ in the region r $\geq a$. (c) Obtain an expression for the current I contained in a circular cross section of radius $r \leq a$ and centered at the cylinder axis. (d) Using Ampere's law, derive an expression for the magnitude of the magnetic field $\overrightarrow{B}$ in the region $r \leq a$. How do your results in parts (b) and (d) compare for $r = a$?
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