Q104 · Practice questionComplex unfamiliar7 marks
QUESTION 104 (7 marks)
An 80-turn rectangular coil of area moves at constant speed across a sharp boundary from a uniform field into the page into a uniform field out of the page. At its leading edge reaches the boundary. It takes to move fully across, and the fraction of its area on the right increases uniformly with time.
a)[2 marks]
Determine the time after when the net external magnetic flux through one turn is zero.
b)[3 marks]
Calculate the magnitude of the induced emf at that instant.
c)[2 marks]
State whether the induced current is clockwise or anticlockwise when viewed from the front of the page. Justify using Lenz's law.
WORKED SOLUTION
7 marksPractice marking scheme
ANSWER
(a) ; (b) ; (c) clockwise
Worked solution
a) Time of zero net flux
Answer: .
Let be the fraction of coil area in the right-hand region. Taking out of the page as positive, the flux through one turn is
At zero net flux,
so
The overlap fraction increases uniformly with time, therefore
b) Induced emf
Answer: .
During the complete crossing, the flux through one turn changes from
to
Therefore
Because the overlap area changes uniformly, the flux changes at a constant rate throughout the interval. Hence the emf has the same magnitude at the zero-flux instant as at every other instant during the crossing:
c) Current direction
Answer: clockwise.
As the coil moves right, the external flux changes from into the page toward out of the page. Lenz's law requires the induced magnetic field to oppose that change, so the induced field is into the page. By the right-hand rule, this requires a clockwise current when viewed from the front of the page.
Alternative approach: use signed flux directly in Faraday's law; the sign of gives the same Lenz-law direction.
Key insight: zero magnetic flux at one instant does not imply zero induced emf; emf depends on the rate of change of flux.
forms the balance between the two oppositely directed flux contributions and obtains .
converts the area fraction to .
determines the total flux change per turn, including the reversal of field direction.
applies Faraday's law with and .
obtains .
states that the induced field must oppose the increasing out-of-page flux, so it is into the page.
identifies the resulting current as clockwise.
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