Question

A uniform magnetic field is perpendicular to the plane of a circular loop of diameter 13 cm formed from wire of diameter 2.6 mm and resistivity of 2.18 × 10-8Ω·m. At what rate must the magnitude of the magnetic field change to induce a 11 A current in the loop?

Answer 1

To induce a current in the circular loop, a changing magnetic field is required. According to Faraday's law of electromagnetic induction, the induced emf (electromotive force) is directly proportional to the rate of change of magnetic flux through the loop. To find the rate at which the magnetic field must change to induce a current of 11 A in the loop, we can use the formula Φ = BA and rearrange it to solve for the rate of change of the magnetic field (dB/dt).

Step-by-step explanation:

To induce a current in the circular loop, a changing magnetic field is required. According to Faraday's law of electromagnetic induction, the induced emf (electromotive force) is directly proportional to the rate of change of magnetic flux through the loop. The magnetic flux through a circular loop of area A is given by Φ = BA, where B is the magnetic field and A is the area of the loop.

In this case, to induce an 11 A current in the loop, we need to find the rate at which the magnetic field must change. We can start by calculating the magnetic flux when the current is 11 A. The area of the loop can be calculated using the formula A = π(r^2), where r is the radius of the loop. Given that the loop has a diameter of 13 cm, the radius is 6.5 cm (= 0.065 m).

Now we can calculate the magnetic flux using the given values. Once we have the magnetic flux, we can rearrange the formula Φ = BA to solve for the rate at which the magnetic field must change (dB/dt) to induce the desired current.

Answer 2

Rate of change of magnetic field is
`3.466* 10^3T/sec`

Step-by-step explanation:

We have given diameter of the circular loop is 13 cm = 0.13 m

So radius of the circular loop
`r=(0.13)/(2)=0.065m`

Length of the circular loop
`L=2\pi r=2* 3.14* 0.065=0.4082m`

Wire is made up of diameter of 2.6 mm

So radius
`r=(2.6)/(2)=1.3mm=0.0013m`

Cross sectional area of wire
`A=\pi r^2=3.14*0.0013^2=5.30* 10^(-6)m^2`

Resistivity of wire
`\rho =2.18* 10^(-8)m`

Resistance of wire
`R=(\rho L)/(A)=(2.18* 10^(-8)* 0.4082)/(5.30* 10^(-6))=1.67* 10^(-3)ohm`

Current is given i = 11 A

So emf
`e=11* 1.67* 10^(-3)=0.0183volt`

Emf induced in the coil is
`e=-(d\Phi )/(dt)=-A(dB)/(dt)`

`0.0183=5.30* 10^(-6)* (dB)/(dt)`

`(dB)/(dt)=3.466* 10^3=T/sec`