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Question

A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm>s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field ori- ented perpendicular to the plane of the loop and with magnitude 0.500 T. (a) Find the emf induced in the loop at the instant when 9.0 s have passed. (b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field. 

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Step 1/4

The initial circumference of the coil `c` is `165cm`The rate of decrease in the circumference `"dc"/"dt"` is `12.0"cm"/s`The magnetic field `B` is `0.500T`

Explanation:

A fundamental tenet of physics called Faraday's Law of Electromagnetic Induction outlines the connection between a shifting magnetic field & the induction of an electromotive force within a circuit or conducting loop.

Step 2/4

(a)

Calculate the rate of change of circumference as follows.

`"dc"/"dt" = (d(2pir))/"dt"`

`"dc"/"dt" = 2pi(dr)/"dt"`

`(dr)/"dt" = 1/(2pi)"dc"/"dt"`

`(dr)/"dt" = 1/(2pi)*12.0`

`(dr)/"dt" = 12/(2pi)"cm"/s`

`(dr)/"dt" = 0.12/(2pi)"cm"/s" "" "...(1)`

Calculate the rate of change of area of circular loop as follows.

`"dA"/"dt" = (d(pir^2))/"dt"`

`"dA"/"dt" = 2pir"dr"/"dt"`

`"dA"/"dt" = 2pi"r"*0.12/(2pi)`

`"dA"/"dt" = "r"*0.12`

`"dA"/"dt" = c/(2pi)*0.12" "" "...(2)`

Explanation:

Here, the circumference `c` is given by `2pir` and the area is given by `pir^2`

Step 3/4

(a), continuation

Write the expression for circumference as follows.

`c = 165-"dc"/"dt"*t`

`c = 165-12*9`

`c = 57cm" "" "...(2)`

Substitute in the formula for change in magnetic flux as follows.

`(dphi)/(dt) = B"dA"/"dt"`

`(dphi)/(dt) = 0.500*c/(2pi)*0.12`

`(dphi)/(dt) = 0.500*0.57/(2pi)*0.12`

`(dphi)/(dt) = 5.44*10^-3mV`

The EMF induced is equal to the change in magnetic flux.

`E = (dphi)/(dt)`

`E = 5.44*10^-3mV`

Explanation:

Here, `mV` stands for mili-volts.

Step 4/4

(b)

Explanation:

The "Right-Hand Thumb Rule" is also known as "Right-Hand Grip Rule"

Final Answer

(a), emf induced in the loop at the instant when `9.0 s` have passed is `5.44*10^-3mV`

(b), clockwise

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