4. Thick Cylinder An infinitely long cylinder with an inner radius $a$ and outer radius $b$ has a linear magnetic susceptibility of $chi_m$. A wire carrying current $I$ runs along its centerline as shown. The magnetic field created by the current $I$ magnetizes the cylinder and the magnetized cylinder creates its own magnetic field. In the problem, you are to find the total magnetic field, $B$, due to the wire and magnetized cylinder. 1. Compute $H(s)$ using $oint vec{H} cdot dvec{l} = I_{free encl}$ 2. Compute $B(s)$ using $H(s)$ and $chi_m$. 3. Compute all bound surface currents, $K_b$ 4. Compute the bound volume current density, $J_b$ You can check your answer for $B$ by using $oint vec{B} cdot dvec{l} = mu_0 I_{encl}$ with $I_{encl}$ being all currents - the current on the wire and the bound surface and volume currents. Note that we did not start with this equation to find $B$ because we didn't know the bound currents.
Added by Ismael M.
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To compute H(s), we use the formula f H - dl = Iyree enc. Since the wire carrying current I runs along the centerline of the cylinder, the magnetic field created by the wire will be in the azimuthal direction. Therefore, we can write H(s) = Hφ(s)φ, where Hφ(s) is Show more…
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Madhur L.
Consider two infinitely long, concentric cylinders, with their axes along the z-axis. The inner solid cylinder has a radius a = 0.08 m and carries a current Ia = 4.5 A directed out of the page. This current is uniformly distributed throughout the cylinder. The outer (hollow) cylinder has a radius b = 0.16 m and carries current Ib = 7.5 A into the page as shown in the figure. 1) What is the magnitude of the magnetic field at point A (y = 0.04 m) due to the current in the two cylinders? |BA| = 5.62 × 10^(-6) T |BA| = 1.12 × 10^(-5) T |BA| = 2.25 × 10^(-5) T |BA| = 3.19 × 10^(-5) T |BA| = 1.5 × 10^(-5) T 2) What is the direction of the magnetic field at point A (y = 0.04 m) due to the current in the two cylinders? -x +y -y 3) What is the magnitude of the magnetic field at point C (y = 0.24 m) due to the current in the two cylinders? |BC| = 2.5 × 10^(-6) T |BC| = 10^(-5) T |BC| = 1.87 × 10^(-6) T
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 $r=a ?$
Ameer S.
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