Slide Wire on a V-Shaped Track: Calculating EMF, Current & Area

In summary, the conversation discusses a slide wire with resistance R moving without friction along a track with v shaped rails meeting at an angle of q. The external magnetic field is constant in magnitude and direction. The principles for calculating the magnitude of the motional electromotive force and the direction of the current associated with it are stated. The equation for the area enclosed in terms of x is also discussed, with a hint to use the equation of the slanted line and the slope in terms of q. The induced electromotive force is found to be a function of both x(t) and v(t). Finally, the algebraic equation for v(t) is requested.
  • #1
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A slide wire (resistance R) travels without friction along the track shown below. It starts from position x0 (non-zero) and velocity v0 , to the right as shown. Assume that the v shaped rails meet at an angle of q. Use the coordinate system shown.



The magnitude of the external magnetic field is constant. The direction of the external magnetic field is also constant (into the page).
a.) State the principle that describes how to calculate the magnitude of the motional electromotive force (this can be an equation).

b.) State the principle that describes how to compute the direction of the current associated with the electromotive force. (words)

c.) Write the equation of the area enclosed in terms of x only. (Hint: What is the equation of the slanted line? Write the slope in terms of q.)

d.) Find the induced electromotive force. This will be a function of both x(t) and v(t).

e.) What is v(t)? (Algebraic equation for v(t).)
 

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  • #3


a.) The principle that describes how to calculate the magnitude of the motional electromotive force is Faraday's Law, which states that the magnitude of the induced electromotive force (EMF) is equal to the rate of change of magnetic flux through a surface.

Mathematically, this can be expressed as:

EMF = -N * dΦ/dt

Where N is the number of turns in the coil and Φ is the magnetic flux through the surface.

b.) The principle that describes how to compute the direction of the current associated with the electromotive force is known as Lenz's Law. This law states that the direction of the induced current will be such that it opposes the change in magnetic flux that caused it.

In other words, the induced current will flow in a direction that creates a magnetic field that opposes the change in the external magnetic field.

c.) The equation of the area enclosed can be written as:

A = (x * tan(q)) * x = x^2 * tan(q)

This equation represents the area of the triangle formed by the slide wire and the two rails.

d.) The induced electromotive force can be calculated using Faraday's Law as:

EMF = -N * dΦ/dt = -N * B * A * v(t)

Where B is the magnitude of the external magnetic field and A is the area enclosed by the slide wire.

e.) The velocity of the slide wire can be expressed as:

v(t) = v0 * cos(q) * t

This equation represents the velocity of the slide wire as it travels along the track at an angle of q.
 
  • #4


a.) The principle that describes how to calculate the magnitude of the motional electromotive force is Faraday's Law of Electromagnetic Induction, which states that the magnitude of the EMF induced in a circuit is equal to the rate of change of magnetic flux through the circuit.

b.) The principle that describes how to compute the direction of the current associated with the electromotive force is the Right-Hand Rule, which states that if the direction of motion of a conductor and the direction of the external magnetic field are known, the direction of the induced current can be determined by pointing the thumb of the right hand in the direction of motion and the fingers in the direction of the magnetic field. The direction of the current is then perpendicular to both the motion and the magnetic field.

c.) The equation of the slanted line can be written as y = mx + b, where m is the slope and b is the y-intercept. In this case, the slope is equal to tan(q), so the equation of the slanted line becomes y = tan(q)x + b. Since the area enclosed is a triangle, the equation for the area can be written as A = 1/2 * base * height. The base of the triangle is x, and the height is the y-coordinate on the slanted line, which is y = tan(q)x + b. Therefore, the equation for the area enclosed is A = 1/2 * x * (tan(q)x + b).

d.) To find the induced electromotive force, we can use the equation from part a: EMF = -N * d(phi)/dt, where N is the number of turns in the circuit and d(phi)/dt is the rate of change of magnetic flux. The magnetic flux through the circuit is given by phi = B * A, where B is the magnitude of the external magnetic field and A is the area enclosed by the circuit. Substituting in the equation from part c, we get phi = B * 1/2 * x * (tan(q)x + b). Taking the derivative with respect to time, we get d(phi)/dt = B * 1/2 * (2x * v + x * tan(q) * v). Substituting this into the equation for EMF, we get EMF = -N * B * 1/2 * (2x * v + x * tan(q) * v) = -N
 

1. How does the V-shaped track affect the calculation of EMF?

The V-shaped track does not directly affect the calculation of EMF. However, it does change the direction of the magnetic field, which can affect the direction of the induced current and the sign of the EMF.

2. What factors affect the magnitude of the induced EMF?

The magnitude of the induced EMF depends on the strength of the magnetic field, the length of the wire, and the velocity of the wire moving through the field. It also depends on the angle between the wire and the magnetic field lines.

3. How is the current in the wire determined from the induced EMF?

The current in the wire is determined by Ohm's Law, which states that current is equal to the induced EMF divided by the resistance of the wire. This means that as the induced EMF increases, the current also increases.

4. What is the relationship between the area of the loop and the induced EMF?

The induced EMF is directly proportional to the area of the loop. This means that as the area of the loop increases, the induced EMF also increases. This relationship is described by Faraday's Law of Induction.

5. Can the direction of the induced current be changed by changing the direction of the magnetic field?

Yes, the direction of the induced current can be changed by changing the direction of the magnetic field. This is because the induced current always flows in a direction that opposes the change in magnetic flux, as described by Lenz's Law. So, changing the direction of the magnetic field will also change the direction of the induced current.

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