Faraday’s Law, Inductance and LR circuits

Physics Objectives AND READINGs

Physics objectives: The complete list of PHYS 151 course objectives is on Angel. This assignment focus on

1. determining the induced electric field, EMF, and/or current due to a changing magnetic flux (Faraday’s Law and Lenz’s law)
2. describing the behavior of RL circuits.

Readings: Knight Chap 33.

Useful Equations & Concepts

The magnetic flux  through an area in a magnetic field   is defined as: .

Faraday’s Law of Induction: If the magnetic flux  through a closed loop C changes with time, an induced electric field  is created. The rate of change of the magnetic flux and the electric field are related by:

The potential difference across an inductor is  where L is the inductance of the coil (inductor). Inductance is measured in units of henries (H). Here are some rules of thumb for RL circuits:

• When you make a change in an RL circuit, initially, an inductor acts to oppose the change in current. A long time later, it acts like an ordinary piece of wire.
• Circuits with inductors resist any changes in current; so, if you throw a switch, the current through an inductor cannot change instantaneously (unless, there is only an inductor in the circuits).

The current in series RL circuit is

Rise of current:                                                               Decay of current:

Where  . ** Be careful, this is only valid for a circuit with an inductor and a resistor in series.

Exercise 1: Faraday’s Law and the Solenoid

The figure below shows the cross-sectional view of an ideal solenoid with n turns per unit length and a radius R₁. At some instant in time, the current through the coil is increasing at a rate  and the instantaneous value of the current is  going clockwise as viewed in this picture.  A point charge +Q is located as shown at a distance R₂ from the axis of the solenoid.

From Ampère’s Law , we can find the magnetic field in an ideal solenoid to be  inside and  outside where is the number of coils of wires per unit of length (. The direction is found by the right hand rule.

In exercise 1, your aim is to use Faraday’s Law to determine the instantaneous force experienced by the charge Q as the current changes.

1. Imagine a circular path of radius R₂ concentric with the solenoid. Determine the instantaneous magnetic flux through the area enclosed by this circle. Calculate the rate of change of this flux (just the magnitude) given that the current changes and increases at a rate .

 

If you look at the figure again, you’ll notice that (aside from the charge +Q outside the solenoid), the entire situation looks exactly the same if you rotate the figure about the center of the solenoid. We say, then, that the problem is rotationally symmetric. We can use this symmetry to argue that

1. the electric field induced by the changing current in the solenoid must be oriented tangent to any circular path concentric with the solenoid and
2. the component of the electric field along our imaginary circular path must be constant all along that circle.
3. So, now that we know from symmetry that the induced electric field must be tangent to our imaginary circle of radius R₂, in which sense is the electric field oriented: clockwise or counter-clockwise? One way to answer this is to consider what would happen if there were an actual conducting wire placed along our imaginary loop. Use Lenz’ law to figure out the direction of the current that would be induced if you did this, and from this, determine the direction of the electric field.

3. Given that the magnitude of the electric field must be the same at every point along this imaginary loop, use the results you have obtained so far to finally determine the instantaneous force on the point charge Q.

Exercise 2: LR circuits

In the circuit shown adjacent, V = 60.0 V, R₁ = 10.0 Ω, R₂ = 20.0 Ω, and L = 3 H.

4. Immediately after the switch is closed (right after current starts flowing), what is the value of the current through R₂? Make sure you justify your answer.  Hint: with this switch closed, the circuit is not a single resistor in series with an inductor. To answer, start by imagining what would happen if the inductor was not there.

 

5. Immediately after the switch is closed, what is the magnitude of the rate of change of current through L? What is the direction (up/down) that the current flows through L.  Justify your answer. Hint: The inductor is in parallel with the and must always have the same voltage.

 

6. A long time after the switch is closed, what is the value of the current through R₂? Justify your answer.

 

7. Once the circuit has reached a steady state (a long time after the switch was initially closed), the switch is reopened. Immediately after the switch is reopened, what is the value of the current through R₂? After the switch is reopened, how much time does it take for the current in R₂ to reach approximately 37% of its initial value? Justify your answers.

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