An infinitely long hollow conducting cylinder with inner radius R/2 and outer radius R carries a uni

Q: Why is the magnetic field zero inside the hollow region?

According to Ampere's Law, the magnetic field along a closed loop depends only on the current enclosed by that loop. Since the hollow region contains no conducting material, it encloses zero current, resulting in zero magnetic field.

Q: How do we calculate the enclosed area in the solid region?

The Amperian loop has a radius $r$. However, the region from $0$ to $R/2$ is hollow. So, the effective area carrying current is the area of the full circle of radius $r$ minus the area of the hollow circle of radius $R/2$, which is $\pi r^2 - \pi(R/2)^2$.

Q: Why isn't the graph a straight line in the solid region?

If the cylinder were completely solid from the center, the enclosed current would be proportional to $r^2$, making $B$ proportional to $r$ (a straight line). But because of the hollow core, the enclosed current has a constant subtracted from it, leading to a $1/r$ term in the magnetic field expression, which curves the graph.

Q: What does a negative second derivative mean physically for the graph?

A negative second derivative means the slope of the graph is decreasing as you move to the right. Visually, this creates a curve that bends downwards, or is "concave down", like the top half of a circle.

Q: Why does the magnetic field decrease outside the cylinder?

Once you are outside the cylinder, the total enclosed current becomes constant. As you move further away, the length of your Amperian loop ($2\pi r$) increases. To keep their product constant, the magnetic field $B$ must decrease proportionally to $1/r$.

Question

An infinitely long hollow conducting cylinder with inner radius $R/2$ and outer radius $R$ carries a uniform current density along its length. The magnitude of the magnetic field, $|\mathbf{B}|$ as a function of the radial distance $r$ from the axis is best represented by

Sigma Percentile · Accepted Answer

### Analyzing the Setup Imagine you are standing inside an infinitely long hollow conducting cylinder. The inner radius is $R/2$ and the outer radius is $R$. A steady, uniform current flows through the solid metallic region between these two boundaries. Our mission is to map out the magnetic field $|\mathbf{B}|$ as we travel from the central axis all the way to infinity. To conquer this, we will wield one of the most elegant tools in electromagnetism: **Ampere's Circuital Law**. It states that the line integral of the magnetic field around any closed loop is directly proportional to the total current enclosed by that loop: $$ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{ ext{enclosed}} $$ ### The Hollow Core ($r < R/2$) Let's start our journey from the central axis, moving outwards but staying within the hollow core ($r < R/2$). If we draw an imaginary circular Amperian loop here, how much current does it trap? Absolutely none! The current is confined to the solid outer shell. Since $I_{ ext{enclosed}} = 0$, Ampere's Law immediately tells us that the magnetic field must also be zero. $$ B = 0 $$ So, the graph must start flat on the zero line from $r = 0$ to $r = R/2$. ### The Solid Shell ($R/2 \le r \le R$) Now, things get interesting. As we push into the solid conducting region, our Amperian loop begins to enclose a fraction of the total current. Let the uniform current density be $J$. The area enclosed by our loop of radius $r$ (excluding the hollow center) is $\pi r^2 - \pi (R/2)^2$. The enclosed current is simply the current density multiplied by this area: $$ I_{ ext{enclosed}} = J \left[ \pi r^2 - \frac{\pi R^2}{4} ight] $$ Plugging this into Ampere's Law, we get: $$ B(2\pi r) = \mu_0 J \pi \left( r^2 - \frac{R^2}{4} ight) $$ $$ B = \frac{\mu_0 J}{2} \left( r - \frac{R^2}{4r} ight) $$ This equation clearly shows that the magnetic field is increasing. But to identify the correct graph, we need to know *how* it increases. Is it a straight line? Does it curve upwards or downwards? Let's consult the derivatives! The first derivative tells us the slope: $$ \frac{dB}{dr} = \frac{\mu_0 J}{2} \left( 1 + \frac{R^2}{4r^2} ight) $$ Since this is positive, the function is strictly increasing. Now, the second derivative reveals the concavity: $$ \frac{d^2B}{dr^2} = - \frac{\mu_0 J R^2}{4r^3} $$ Because the second derivative is strictly negative, the curve is **concave down**. It bulges upwards like a dome, starting steep and gradually flattening out as it approaches $r = R$. ### The Outside World ($r > R$) Finally, we emerge outside the cylinder. Now, no matter how large we make our Amperian loop, the enclosed current is simply the constant total current $I_{ ext{total}}$. Ampere's Law simplifies beautifully: $$ B(2\pi r) = \mu_0 I_{ ext{total}} $$ $$ B = \frac{\mu_0 I_{ ext{total}}}{2\pi r} $$ In this region, the magnetic field is inversely proportional to the distance $r$. It decays smoothly towards zero as we move further away. ### The Final Verdict Piecing our journey together: 1. Flat at zero from $0$ to $R/2$. 2. Increasing with a **concave down** curve from $R/2$ to $R$. 3. Decreasing as $1/r$ for $r > R$. When we examine the given options, the only graph that faithfully captures this entire physical reality—especially the crucial concave down shape in the middle region—is option (d).

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Visualized Solution

$The Hollow Cylinder$

$Ampere’s Circuital Law$

$Region 1: r < R /2$

$Region 2: R /2 \leq r \leq R$

$Magnetic Field in Region 2$

$Concavity of the Curve$

$Region 3: r > R$

$Conclusion$

The Sigma Insight: Ampere's Circuital Law

Analyzing the Setup Imagine you are standing inside an infinitely long hollow conducting cylinder

The Hollow Core ( $r < R /2$ ) Let's start our journey from the central axis, moving outwards but staying within the hollow core ( $r < R /2$ )

The Solid Shell ( $R /2 \leq r \leq R$ ) Now, things get interesting

The Outside World ( $r > R$ ) Finally, we emerge outside the cylinder

The Final Verdict Piecing our journey together: 1

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Hollow Cylinder

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ampere’s Circuital Law

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Region 1: r<R/2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Region 2: R/2≤r≤R

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Magnetic Field in Region 2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Concavity of the Curve

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Region 3: r>R

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion