PHYS 151 University of Engineering & Technology Lahore Faradays Law Worksheet

  

This HW is about INDUCTANCE AND LR CIRCUITS, and you can just follow instructions in the word file and type answers in the file.

FARADAY’S LAW, INDUCTANCE AND LR CIRCUITS
PHYSICS OBJECTIVES AND READINGS
Names:
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 𝛷𝑚 through an area in a magnetic field 𝐵
⃗ ⋅ 𝑑𝐴.
is defined as: 𝛷𝑚 = ∫ 𝐵
Faraday’s Law of Induction: If the magnetic flux 𝛷𝑚 through a
closed loop C changes with time, an induced electric field 𝐸⃗ is
created. The rate of change of the magnetic flux and the
electric field are related by:
∮ 𝐸⃗ ⋅ 𝑑𝑠 = −
The potential difference across an inductor is 𝛥𝑉𝐿 = −𝐿
Section #______________
Date: _____________
𝑑𝛷𝑚
𝑑𝑡
𝑑𝐼
𝑑𝑡
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:
o
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.
o
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:
𝐿
ℇ
𝐼 = 𝑅 (1 − 𝑒
−
𝑡
𝜏𝐿
)
Decay of current:
𝐼 = 𝐼0 𝑒
−
Where 𝜏𝐿 = 𝑅 . ** Be careful, this is only valid for a circuit with an inductor and a resistor in series.
𝑡
𝜏𝐿
2
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
𝑑𝐼
radius R1. At some instant in time, the current through the coil is increasing at a rate 𝑑𝑡 and the
instantaneous value of the current is 𝐼 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.
Looking this way
⃗ ⋅ 𝑑𝑠 = 𝜇0 𝐼𝑡ℎ𝑟𝑜𝑢𝑔ℎ , we can find the magnetic field in an ideal solenoid to be 𝐵 =
From Ampère’s Law ∮ 𝐵
𝜇0 𝑛𝐼 inside and 𝐵 = 0 outside where 𝑛 is the number of coils of wires per unit of length (𝑛 = 𝑁/𝐿). 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
𝑑𝐼
(just the magnitude) given that the current changes and increases at a rate 𝑑𝑡.
3
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
i)
ii)
the electric field induced by the changing current in the solenoid must be oriented tangent to
any circular path concentric with the solenoid and
the component of the electric field along our imaginary circular path must be constant all
along that circle.
2. 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 counterclockwise? 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.
4
R1
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.
+
V
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.
−
R2
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 𝑅2 and must always have the same voltage.
L
5
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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