Ehan 09 Apr 2016 - 2

Question

15) A magnetic field B is applied through a coil in a direction perpendicular to the plane of the coil as shown.

Use Faraday's law and Lenz's law to state and explain the direction of the induced current through resistor R when the magnetic field B is decreasing. [4]

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Answer

When the magnetic flux density B decreases, the magnetic flux linkage of the coil into the page, given by the formula ϕ = B A, decreases. By Faraday's Law of electromagnetic induction, there will be an induced e.m.f. produced in the coil.

This induced e.m.f. in the coil will cause an induced current to flow.

By Lenz's law, since the magnetic flux linkage into the page has deceased, the induced current will produce magnetic flux into the page. According to the right hand grip rule, the induced current will flow in the clockwise direction as shown below.

Therefore, the current will flow through the resistor from the right to the left.

Ehan 09 Apr 2016 - 1

Question

12) A long bar magnet is suspended from a helical spring so that one pole of the magnet lies within a long cylindrical coil as shown.

The magnet is given a small vertical displacement so that one pole of the magnet oscillates within the coil.

a) An induced e.m.f. is measured across the terminals of the coil.

i) Sketch the graph of e.m.f. against time to show how the induced e.m.f. varies with time for two periods of oscillation. [2]

ii) Using the principles of EM induction, explain the variation of the induced e.m.f. shown in your graph. [3]

b) State and explain a change that occurs in the amplitude of the oscillations of the magnet when a resistor is connected across XY. [3]

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Answer

a) i) Assuming that the magnet is displacement upwards and then released, the induced e.m.f. is as shown:

a) ii) At the top-most position, the magnetic flux linkage is the least. This because the magnetic field lines of the bar magnet spread out so the fewer magnetic field lines pass through the bottom-most coil.

At the bottom-most position, the magnetic flux linkage is the greatest as most of the magnetic field lines pass through the bottom-most coil.

Assuming that the magnet is displaced upwards and released, the magnetic flux linkage varies as shown:

By using the relationship \( \varepsilon = -{{d \phi } \over {d t}}\) , the graph of the induced e.m.f. against time is obtained.

b) When a resistor is connected across XY, the induced e.m.f. will cause an induced current to flow.

According to Lenz's law, the induced current flowing in the coil will produce a magnetic field to oppose the change producing it. In this case, the magnetic field produced by the induced current will repel the magnet as it falls into the coil and attract the magnet as it rises away from the coil.

This causes the amplitude of the oscillation of magnetic to decrease with time. The kinetic energy of the magnet is converted into electrical energy which is then dissipated as thermal energy by the resistor.

Minqi 02 Apr 2016 - 2

Question

The figure below shows the cross section of a long cylindrical magnet, as well as the plan view of the radial magnetic field. The cylindrical magnets are placed vertically. A thin circular aluminum ring of radius r and resistance R falls through the magnetic field as shown in the diagram. In the vicinity of the aluminum ring, themagnetic flux density is 2.0 T.

[Resistivity of aluminium = 3.1 x 10^-8 Ω m, density of aluminium = 2.7 x 10³ kg m^-3]

a) By considering the plan view of the magnet, indicate the polarity of the magnets and the direction of the induced current.

b) Describe the motion of the aluminum ring when it is released in the radial magnetic field.

c) Show that the current in the aluminum ring when it is falling at a speed v is given by

\( I= {{2 \pi r B v} \over {R}}\)

d) Calculate the terminal velocity of the ring.

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Answer

a) Magnetic field lines are from the North pole to the South pole, hence the outer ring is a North pole and the inner cylinder is a South pole.

Based on Fleming's Right hand rule, since the ring is moving into the page and the magnetic field lines are radially inwards, the induced current flows in the anti-clockwise direction.

b) When it is first released, it falls with an acceleration of 9.81 ms^-2. As velocity of the the aluminium ring increases, the acceleration of the ring decreases until it eventually reaches zero. The ring has reached terminal velocity.

c)

d)

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Minqi 02 Apr 2016 - 1

Question

A long, straight wire, which carries a current, is placed in the vicinity of a circular coil. Using the definition of magnetic flux, explain the following statements.

a) When the wire is placed along the diameter of the coil, the net flux through the coil is zero.

b) When the wire is placed along the axis to the coil, the net flux through the coil is zero.

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Answer

a) The perpendicular component of the magnetic flux density with respect to the area of the coil on the left side of the coil is into the page.

The perpendicular component of the magnetic flux density with respect to the area of the coil on the right side of the coil is out of the page.

By symmetry, the magnitude of the magnetic flux density on both sides are the same.

Hence, the resultant perpendicular component of the magnetic flux density is zero. Therefore, the magnetic flux through the coil is zero.

b) The magnetic field lines are along the surface of the coil. This means that the perpendicular component of the magnetic flux density with respect to the area of the coil is zero.

Hence, the magnetic flux through the coil is zero.