Question

If a long hollow copper pipe carries a direct current, the magnetic field associated with the current will be (a) Only inside the pipe (b) Only outside the pipe (c) Neither inside nor outside the pipe (d) Both inside and outside the pipe

Short answer

Using Ampere's law, we found that the magnetic field is zero inside the hollow copper pipe when the radius is less than the pipe's inner radius and nonzero outside the pipe when the radius is greater than the pipe's outer radius. Therefore, the correct answer is (b) Only outside the pipe.

Step-by-step solution

Recall Ampere's Law

Ampere's law states that the closed-line integral of the magnetic field (B) around a closed loop is equal to the product of the permeability constant (μ₀) and the total current (I) passing through the loop. Mathematically, the law can be represented as follows: \[\oint_{loop} \vec{B} \cdot d\vec{l} = \mu_0 I\] Keep in mind that this law will be useful to analyze the magnetic field distribution for the given hollow copper pipe.

Consider a long hollow copper pipe

Let's consider a long hollow copper pipe with direct current, I, flowing through it. In this problem, we need to determine at which region the magnetic field is present (inside the pipe, outside the pipe, or both).

Analyze the magnetic field inside and outside the pipe

For understanding the distribution of the magnetic field, we can choose two Amperian loops: (a) An Amperian loop inside the Copper pipe (radius < pipe's inner radius): Since there is no current passing through this loop, according to Ampere's law, the magnetic field inside the pipe (when radius < pipe's inner radius) will be zero. (b) An Amperian loop outside the Copper pipe (radius > pipe's outer radius): Now, the total current (I) is enclosed by this loop. Applying Ampere's law to this loop, we find that the magnetic field outside the pipe (when radius > pipe's outer radius) is nonzero. From the analysis above, we can conclude the distribution of the magnetic field associated with the direct current flowing through the long hollow copper pipe.

Choose the correct option

After analyzing the magnetic field using Ampere's law, we can now choose the correct option: (a) Only inside the pipe -> This option is incorrect, as we found that the magnetic field is zero inside the pipe. (b) Only outside the pipe -> This option is correct, as we found that the magnetic field is nonzero outside the pipe. (c) Neither inside nor outside the pipe -> This option is incorrect, as we found that there is a magnetic field outside the pipe. (d) Both inside and outside the pipe -> This option is incorrect, as we found that the magnetic field is zero inside the pipe but nonzero outside the pipe. So, the correct answer is (b) Only outside the pipe.

Key concepts

Magnetic Field

A magnetic field is an invisible force field that is created by moving electric charges, like direct current. It exerts forces on other moving charges and magnetic materials. Magnetic fields are essential in various applications, such as motors, generators, and magnetic resonance imaging (MRI).

The strength and direction of a magnetic field at a given point are described by magnetic field lines. These lines never cross each other, and they always form closed loops. When analyzing where a magnetic field exists, it's critical to use concepts like Ampere's Law as it relates to direct current and loops.

Magnetic fields are generated by:

• Electric currents (permanent magnets or circulating charges)
• Changing electric fields (time-varying electric fields)

Understanding magnetic fields is key when considering the forces acting on objects due to direct currents, especially in practical applications such as in this problem with the hollow copper pipe.

Hollow Copper Pipe

A hollow copper pipe has a unique structure, with a void in its center and a conducting surface. Copper is often used because of its excellent electrical conductivity, resistance to corrosion, and cost-effectiveness. When current flows through the material of the pipe, it influences the magnetic field around it.

In the case of a hollow pipe, it's crucial to understand that the current only flows through the material of the pipe. Inside the hollow section, there's no conducting material to allow a current pathway. This characteristic is vital in understanding why, according to Ampere's Law, no magnetic field exists within the pipe.

Key properties of copper that affect its usage and the behavior of currents include:

• High electrical conductivity
• Low resistivity
• Good thermal conductor

These properties make copper an ideal choice for scenarios involving direct current, as illustrated in this exercise with the absence of a magnetic field inside the hollow section.

Direct Current

Direct current (DC) refers to the unidirectional flow of electric charge, often from a battery or DC power supply. Unlike alternating current (AC), which changes direction periodically, DC maintains a constant direction and amplitude. This steady flow makes it easier to analyze magnetic fields and currents using Ampere’s Law.

In this problem, the direct current flows through the walls of the hollow copper pipe, impacting the surrounding magnetic field. The consistency of direction in direct current simplifies calculations of magnetic fields using closed-loop integrals around the pipe.

Key features of direct current include:

• Stable voltage and current direction
• Used in electronic devices, transportation (like electric vehicles), and more
• Allows for the straightforward application of laws like Ampere's law to determine magnetic forces

Direct current is vital when considering electromagnetic interactions like those in the hollow copper pipe since it leads to predictable and stable magnetic effects, helping to better understand concepts laid out in step-by-step solutions.

Amperian Loop

An Amperian loop is an imaginary closed loop used primarily in conjunction with Ampere’s Law to calculate the magnetic field. It is an essential tool in electromagnetism, helping to determine where magnetic fields are present around current-carrying conductors.

Choosing the right Amperian loop is crucial when analyzing the distribution of magnetic fields in different scenarios, like with a hollow copper pipe. To apply Ampere's Law, the loop can be conceptualized inside and outside the conductor.

Key points to understand about Amperian loops include:

• Form a closed path around the conductive area to use Ampere's Law effectively
• Help define the boundary for evaluating magnetic forces
• The loop's positioning affects whether it encloses the current and impacts the calculated magnetic field

In the hollow copper pipe context, two Amperian loops are considered: one inside the pipe (enclosing no current) and one outside (enclosing the entire current). This helps conclude that the magnetic field exists only outside the pipe, illustrating a clear application of Ampere's Law.