The magnetic flux through a wire loop is plotted in Figure...

The magnetic flux through a wire loop is plotted in Figure $23.25$. (a) Is the magnitude of the induced emf at $t=0.25 \mathrm{~s}$ greater than, less than, or the same as the magnitude of the induced emf at $t=0.55 \mathrm{~s}$ ? Explain. (b) Calculate the magnitude of the induced emf at $t=0.25 \mathrm{~s}$ and $t=0.55 \mathrm{~s}$.

Key Concepts

Induced Electromotive Force (emf)

Induced emf is the voltage generated in a circuit due to time-varying magnetic flux. It is a central concept in electromagnetism and is calculated by taking the negative time derivative of the magnetic flux, indicating that it is not the absolute value of the flux but its change over time that matters.

Rate of Change of Magnetic Flux

The rate of change of magnetic flux, or the derivative of the flux with respect to time, determines the magnitude of the induced emf according to Faraday's law. A larger rate of change (a steeper slope on a flux versus time graph) results in a higher induced emf, regardless of the absolute values of the magnetic flux.

Magnetic Flux

Magnetic flux is a measure of the quantity of magnetic field lines passing through a given surface area. It is fundamental in understanding electromagnetic induction because changes in magnetic flux over time are what drive the induction of an electromotive force (emf) in a loop.

Faraday's Law of Electromagnetic Induction

Faraday's law states that any change in the magnetic environment of a coil of wire will cause an emf to be induced in the coil. Specifically, the magnitude of the induced emf is equal to the absolute rate of change of the magnetic flux through the loop. This principle is essential for calculating induced emfs in various scenarios.

Transcript

00:01 Hello and welcome to this video solution of numerate.

00:04 Here the magnetic flux through a wire loop is plotted in this figure 23 .25 a is the magnitude of the induced emf at equal to 0 .25 greater than or same as that of the magnitude of induced tmf at 0 .55 seconds you have to do right so what i can do is you can just draw i can just draw a simple diagram of this this graph so here the flux phi is plotted along the y axis and you have got time along the x axis right so it's something like this right and this is point one next here we have point two this point where it cuts let's say this is point four six let's say this point is somewhat 0 .6 and this is 0 .9 right so this is what it's given and let's say this is 10 weber and on the other side this is negative 5 weber right so this is the complete graph it's given so first of all you have to find out the induced mf at equal to 25 seconds right so 25 seconds is somewhat this curve right so what you can do is you need to find out the slope of this because this induced mf epsilon is equal to d5 delta 5 over delta t right so delta 5 what we can do is because this slope is constant so here you have got delta 5 that is you need to have two points so the final emf flux is negative 5 sorry induced mf is a negative defided t fine so negative of here we have minus 5 minus of 10 because 10 is initial and the difference in time is when the final time is 0 .6 minus initial time is 0 .2 right and this is given by 5 sorry 15 over 0 .4 volts right it's 37 .5 is the induced amf right right is the amf right is the emf.

03:25 Next at t equal to 55 seconds you have to tell so t equal to 55 seconds is below 0 .6 right so obviously emf at t equal to 55 seconds is also same because the slope is same there also right so we can say that it's equal right for emf at 0 .25 seconds that is equal to emf at 0 .55 seconds right so so this is what you have.

04:14 0 .55...

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