Question

Learning Goal: To understandAmpère’s law and its application.

Ampère’s law is often written $\oint \vec{B}(\vec{r}) \cdot d\vec{l} = \mu_0 I_{\rm encl}$ .

Part A

The integral on the left is

a) theintegral throughout the chosen volume.
b) the surface integral over theopen surface.

c) thesurface integral over the closed surface bounded by theloop.

d) theline integral along the closed loop.

e) theline integral from start to finish.

Part C

The circle on the integral means that $\vec{B}(\vec{r})$ must be integrated

a) over acircle or a sphere.

b) alongany closed line that you choose.

c) alongthe path of a closed physical conductor.
d) over the surface bounded by thecurrent-carrying wire.

Part D

Which of the following choices of path allowyou to use Ampère’s law to find $\vec{B}(\vec{r})$ ?

1. The path must pass through the point $r_vec$ .
2. The path must have enough symmetry so that $\vec{B}(\vec{r}) \cdot d\vec{l}$ is constant along large parts of it.
3. The path must be a circle.

a)  aonly
b) a and b

c) a and c

d) b and c

Part E

Ampère’s law can be used to findthe magnetic field around a straight current-carrying wire.

Is this statement true orfalse?

Part F

Ampère’s law can be used to findthe magnetic field at the center of a square loop carrying aconstant current.

Is this statement true orfalse?

Part G

Ampère’s law can be used to findthe magnetic field at the center of a circle formed by acurrent-carrying conductor.

Is this statement true orfalse?

Part H

Ampère’s law can be used to findthe magnetic field inside a toroid. (A toroid is a doughnut shapewound uniformly with many turns of wire.)

Is this statement true orfalse?

Answer 1

Concepts and reason

The concept required to solve this problem is Ampere’s Circuital law.

Use the Ampere circuital law to identify the correct statements.

Fundamentals

The Ampere-Circuital law gives the line integral of a magnetic field along a closed loop is the product of current enclosed and permeability of the material.

The expression of the Ampere-Circuital law is expressed as follows:

$\oint {\vec B\left( {\vec r} \right) \cdot d\vec l = {\mu _0}{I_{{\rm{encl}}}}}$

Here, $\vec B\left( {\vec r} \right)$ is the magnetic field, $\vec l$ is the length vector, ${\mu _0}$ is the permeability of vacuum, and ${I_{{\rm{encl}}}}$ is the total current enclosed in the loop.

(A)

On the left side in the ampere’s law the integral is over $d\vec l$ which is length vector. The circle on the integral symbol means the integral is closed that is the integral is over a closed loop.

The integral in not throughout the chosen volume, but only through the closed loop. It is single integral which implies it is only line integral and not surface integral. The line integral from start to finish may not be a closed loop. However, for the Ampere law it is must to have a closed loop.

Therefore, correct statement is that the integral on the left is the line integral along the closed loop.

(C)

The circle on the integral symbol means the integral is closed that is the integral is over a closed loop. The shape of loop can be arbitrary and may not be a circle. The path is arbitrary and may not be along a closed physical conductor. The integral is over line or length and not surface.

Therefore, it is not over the surface bounded by the current-carrying wire. Hence, the circle on the integral means that $\vec B\left( {\vec r} \right)$ must be integrated along any closed line that you choose.

(D)

The magnetic field in the Ampere’s law should be on the line of integration that is the path chosen must pass through point$\vec r$. If the path chosen does not pass through, $\vec r$ then the value of magnetic field used in integration is wrong and left side may not match with the right side of the Ampere’s law. So, it is must that path pass through the point $\vec r$.

The magnetic field must be enough symmetric that integration gives a correct result. The discrete quantitates cannot be integrated directly. So, the path must have enough symmetry so that $\vec B\left( {\vec r} \right) \cdot d\vec l$ is constant along large parts of it. Otherwise integration will not give correct result. The path is of any arbitrary shape. It may not be a circle. Hence, only statement a and b are correct.

(E)

For a straight current carrying wire, a symmetric closed path can be constructed in the plane of cross section of wire, which encloses a current, and magnetic field is also perpendicular to the circumference of this circular ring around the wire inside or outside the wire.

Thus, Ampere’s law can be used to find the magnetic field around a straight current-carrying wire.

(F)

The square closed loop does not have symmetric magnetic field for any closed path passing through its center. The path through center for which magnetic field is symmetric does not enclose any current.

Therefore, it is not possible to find the magnetic field at the center of a square loop carrying a constant current by using Ampere’s law.

(G)

The circular closed loop does not have symmetric magnetic field for any closed path passing through its center. The path through center for which magnetic field is symmetric does not enclose any current.

Therefore, it is not possible to find the magnetic field at the center of a circle formed by a current carrying conductor by using Ampere’s law.

(H)

A toroid is a doughnut shaped with wound wire making the surface of the doughnut. The magnetic field inside a toroid is constant for a circular loop inside it. The current enclosed in such a loop is also not zero.

The symmetric value of current with a line passing through the point inside the toroid satisfies the necessary condition for Ampere’s law.

Thus, Ampere’s law can be used to find the magnetic field inside a toroid.

Ans: Part A

The integral on the left is the line integral over a closed loop.

Part C

The circle on the integral means that $\vec B\left( {\vec r} \right)$ must be integrated over along any closed line that you choose.

Part D

The correct statements are a and b.

Part E

Ampere’s law can be used to find the magnetic field around a straight current-carrying wire is True.

Part F

Ampere’s law can be used to find the magnetic field at the center of a square loop carrying a constant current is False.

Part G

Ampere’s law can be used to find the magnetic field at the center of a circle formed by a current carrying conductor is False.

Part H

Ampere’s law can be used to find the magnetic field inside a toroid is True.