The magnetic flux through a coil is given by ΦB=αt-βt^3 where αand βare constants. (a) What are

The magnetic flux through a coil is given by ΦB=αt-βt^3...

The magnetic flux through a coil is given by $\Phi_{B}=\alpha t-\beta t^{3}$ where $\alpha$ and $\beta$ are constants. (a) What are the units of $\alpha$ and $\beta ?$ (b) If the induced emf is zero at $t=0.500 \mathrm{~s},$ how is $\alpha$ related to $\beta ?$ (c) If the emf at $t=0$ is $-1.60 \mathrm{~V},$ what is the $\mathrm{emf}$ at $t=0.250 \mathrm{~s} ?$

Key Concepts

Faraday's Law of Electromagnetic Induction

This law states that an electromotive force (emf) is induced in a coil when there is a change in the magnetic flux through it over time. The magnitude of the induced emf is equal to the negative rate of change of the magnetic flux. Understanding this law allows one to relate the time derivative of magnetic flux to the induced voltage in the coil.

Unit Analysis in Physics

Unit analysis involves ensuring that both sides of an equation have consistent dimensions. When dealing with time-dependent functions, such as a magnetic flux expressed in Weber (Wb) as a function of time, determining the units of coefficients requires equating them according to their dimensional contributions. This concept is critical for identifying the physical dimensions of constants in a given equation.

Differentiation of Time-dependent Functions

Differentiation is the mathematical process used to find the rate at which a quantity changes with respect to time. In the context of magnetic flux, differentiating the flux function with respect to time yields the induced emf. This concept is essential for connecting how changes in flux affect the induced electrical effects in a coil.

Solving for Relationships Between Constants

When additional conditions are provided (such as specifying particular values of time or induced emf), it becomes necessary to substitute these conditions into the differentiated equation. This process allows for the determination of relationships between the constants in the expression, ensuring that the physical constraints of the problem are met.

Transcript

00:01 Hi, in the given problem, magnetic flux linked through the coil is given as alpha t minus beta tq means it is varying as a function of time.

00:17 In the first part of the problem, we have to find the unit for alpha and unit for beta.

00:23 So for the unit associated with alpha, as we know, using principle of homogeneity.

00:36 All the terms in an equation should have same dimensional equation.

00:43 So here this 5b is weber.

00:46 So this alpha t should also be having the si unit as vapor.

00:53 This alpha will become vapor per second which is the si unit of time.

01:01 So the unit of alpha will be written as vapor per second.

01:09 Similarly for beta this beta t cube should also have the units of magnetic flux means weber so this beta will come out to be weber divided by cube of time or we can say this is weber per second cube so these are the two answers for the first part of this problem.

01:39 Now, in the second part of the problem, we have to find the value of alpha in terms of beta for which it is given that emf induced is 0 at the time t is equal to 0 .5 0 second.

02:03 So first of all, we will find an expression for emf induced using para is laws of electromagnetic induction, which says emf induced is actually given by the time derivative of the function of magnetic flux.

02:29 So it becomes minus d by dt of alpha t minus beta t cube.

02:38 So doing the differentiation, it becomes minus alpha minus 3 beta p square and it is given as 0 at the time of 0 .50 second.

02:57 So this is alpha which we have to find minus 3 beta which is unknown and for time we use 0 .5 0 .50 square.

03:09 So here it becomes alpha is equal to 3 times of beta.

03:16 Into the square of 0 .5, which comes out to be 0 .25.

03:22 So finally, this is 0 .75 beta.

03:27 Hence, we can write this alpha either as 0 .75 beta or as 3 fourth of beta...