Feb 2024 Update: A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at

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.

Asked By StormySkies20 at 01 Feb 2024

Answered By Expert

Anthony

Expert · 2.1k answers · 2k people helped

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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