5.7.3: Problems

Exercise \(\PageIndex{1}\): Force between wires

The animation shows wires coming out of the screen. For each configuration, how many wires have current flowing in the same direction as the red wire? You can drag the red wire as well as the other wires. Explain. Restart.

Problem authored by Mario Belloni and modified by Anne J. Cox.

Exercise \(\PageIndex{2}\): Adding magnetic fields from wires

The current in the two red wires is \(3\text{ A}\) coming out of the page (position is given in meters). The black wire has a current of zero right now, but can have current either into or out of the page depending on the value of the slider. Restart.

1. Where does the black wire need to be placed (you can drag it around), and what current does it need to have so that the magnetic field will be zero at the origin \((0\text{ m},\: 0\text{ m})\)?
2. Develop an expression for the position of the black wire as a function of current so that the magnetic field will be zero at the origin. Verify your expression with the animation.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{3}\): Find unknown currents for a given path integral

Find the current carried by each of the four wires in the animation. Each wire carries current either into or out of the computer screen (position is given in millimeters and the magnetic field strength is in given in millitesla, \(10^{-3}\text{ T}\), so the integral is given in \(\text{mT}\cdot\text{mm}=10^{-6}\text{ T}\cdot\text{m}\)). You can turn the integral on, and the cursor will change to a pencil and draw the path as it calculates the integral of the magnetic field in the direction of the path you take (path integral of \(\mathbf{B}\cdot d\mathbf{l}\).) You can re-zero the integral at any point or turn the integral off and move the cursor to another spot. Restart.

Problem authored by Anne J. Cox.
Script authored by Mario Belloni and Wolfgang Christian and modified by Anne J. Cox.

Exercise \(\PageIndex{4}\): Amperian loop and path integral for uniform field

1. Find \(\int\mathbf{B}\cdot d\mathbf{l}\) for each side of the square wire (position is given in millimeters and the magnetic field strength is given in millitesla, \(10^{-3}\text{ T}\)).
2. What is the total value of the path integral \(\int\mathbf{B}\cdot d\mathbf{l}\) for the field shown? Explain.

Restart.

Problem authored by Anne J. Cox

Exercise \(\PageIndex{5}\): Coaxial cable

A coaxial cable consists of an outer conductor and an inner conductor separated by an insulating plastic filler. These two conductors usually carry equal currents in opposite directions. You may have seen this type of configuration on cable TV hookups as well as on certain types of computer networks. The animation shows the progression of the magnetic fields as many small wires are added to produce a coaxial configuration (position is given in millimeters and magnetic field strength is given in microtesla, \(10^{-6}\text{ T}\)). Restart.

1. Build the coaxial cable by adding the current-carrying wires. Explain why this type of cable might be preferable to household wiring or lamp cord which consists of two side-by-side wires carrying currents in opposite directions.
2. Explain why the field is zero outside the cable and within the center (blue) cable.
3. Click-drag to measure the field at any point. Find the current passing through the inner conductor.

Problem authored by Melissa Dancy, Mario Belloni and Wolfgang Christian.

Exercise \(\PageIndex{6}\): Force between wire and cylinder

Find the current in the red wire. You can drag the red wire around and read the force/length on it. You can also read the value of the magnetic field at given points in the animation when your mouse is not on a wire (position is given in centimeters, magnetic field strength is given in \(10^{-5}\) tesla, and force/length is given in newtons/meter). You can also turn the current in the red wire on and off by pushing the buttons. Restart.

Problem authored by Anne J. Cox.
Script authored by Wolfgang Christian and Mario Belloni and modified by Anne J. Cox.

Exercise \(\PageIndex{7}\): Wire with uniform current

The gray circle in the center represents a cross-section of a wire carrying current coming out of the computer screen (position is given in millimeters, \(10^{-3}\text{ m}\), and magnetic field strength is given in millitesla, \(10^{-3}\text{ T}\)). The current is uniformly distributed through the wire. The black circle is an Amperian loop with a radius you can change with the slider. Restart.

1. What is the direction of the current in the wire?
2. What is the current in the wire?

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{8}\): Current carrying plate

When you push the "add current" button, current-carrying wires are added to a region of space with a uniform magnetic field to form a plate carrying current either out of the screen or into the screen (position is given in centimeters and the magnetic field strength is given in millitesla). Restart.

1. Before you add the current, draw an Amperian path that makes use of the symmetry of the field. What is the value of the path integral?
2. Add the current. Draw a path that encloses a number of wires and use this path to find the current per unit length along the \(x\) axis (i.e., if you think of the plate as made up of the wires you see in cross section, how much current is carried on the wires from \(x = 0\text{ cm}\) to \(x = 1\text{ cm}\)?). Ignore edge effects.

Problem authored by Melissa Dancy and Wolfgang Christian.

Exercise \(\PageIndex{9}\): Slinky Solenoid

A solenoid is made by wrapping a long wire many times around a cylinder. This animation shows a cross section of the solenoid (cylinder). Each loop of the wire circles behind and in front of the computer screen, so your view of the solenoid is a long tube sliced in half, lying on its side (position is given in centimeters and magnetic field is given in millitesla). Use the slider to change the current through the wire. This solenoid has a fixed number of wire coils, but by using the slider, you can stretch it or compress it (think of a solenoid made from a Slinky^®, in which the ends of the Slinky^® are connected to a current source). Restart.

1. The black box is an Amperian loop. For which sides of the box is \(\int\mathbf{B}\cdot d\mathbf{l}=0\)? Why? For the other side, show that \(\int\mathbf{B}\cdot d\mathbf{l}=BL\), where \(L\) is the length of the side and \(B\) is the magnitude of the magnetic field at that point. How much current is enclosed in the Amperian loop?
2. Therefore, how many loops/centimeter are there?
3. How many total loops are there in this solenoid (how many coils in the Slinky^®)?
4. Develop an expression for the magnetic field as a function of the length of the solenoid (same number of loops as the solenoid is stretched or compressed).

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{10}\): Magnetic field of a loop

The animation shows the cross section of a loop of wire oriented perpendicular to the screen. It carries a current into and out of the screen (position is given in millimeters and magnetic field is given in millitesla). You can move the loop (drag at the middle) or change its radius (drag either the red or blue dot). Restart.

1. What is the current in the loop?
2. Develop an expression for the magnetic field in the center of the loop as a function of the radius of the loop. Verify your expression with the animation.

Problem authored by Mario Belloni and modified by Anne J. Cox.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.