Faraday's law with Calc 3 integration help

In summary, the problem involves finding the induced voltage and current of an equilateral triangular circuit with resistance R, which is located a distance a away from an infinitely long wire carrying current I=sin(ωt). The magnetic flux is determined using Faraday's law and Ohm's law is also relevant. The process involves finding the magnetic field produced by the wire, then integrating over the area of the triangle to find the magnetic flux. A partial check can be done by evaluating the integral of ydx to see if the correct area is obtained. The result for the flux is then plugged into Faraday's law to find the induced voltage.
  • #1
horsewnoname

Homework Statement


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Suppose an infinitely long wire carrying current ##I=sin_0(\omega t)## is a distance ##a## away from a equilateral triangular circuit with resistance ##R## in the same plane as shown in the figure. Each side of the circuit is length ##b##. I need to find the induced voltage ##V_{ind}## and current ##i## of the triangular circuit.

Homework Equations


Faraday's law ##V_{ind}=-\frac{\mathrm{d} \phi}{\mathrm{d} x}## where magnetic flux ##\phi = \int \boldsymbol{B}\cdot d\boldsymbol{s}## and ##d\boldsymbol{s}## is the surface through which the flux intercepts.

Also relevant is Ohm's law which is simply ##V=IR##.

The Attempt at a Solution


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So the first thing I have to do is determine the ##\boldsymbol{B}## produced by the infinite wire. This is simply ##\boldsymbol{B}=\frac{\mu_0 I}{2 \pi x}##.

From this, I can determine the magnetic flux which is the tricky part. To do this, I need to integrate over the area of the triangle which means I need a double integral where the horizontal ##x## limits of integration are given by ##x=a## to ##x=a+\frac{\sqrt3}{2}b## and the vertical ##y## limits of integration are given by ##y=0## to ##y=\frac{-1}{\sqrt3}x+\frac{a}{\sqrt3}+\frac{b}{2}##. I got this equation of a line by splitting the triangle in half as I show in the figure above. The only thing is that I will have to multiply the result by 2 to consider both areas of the triangle, not just half.

It's been a while since I've taken Calc 3, but I eventually got to the result ##\frac{\mu_0I}{\pi} [\frac{-b}{2}+(\frac{a}{\sqrt3}+\frac{b}{2}) ln(\frac{a+\frac{\sqrt3 b}{2}}{a})]##.

From here, I know I can just plug ##\phi## into Faraday's law and get the induced voltage, but I don't know if I did the integral correctly. Is there any way for me to check that I have the correct result?
 
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  • #2
A partial check to see if your expression for y and your limits are correct is to see if you get the correct area for the triangle if you evaluate ∫ydx.

Your explanation of how you set up the integral sounds right. I think your answer for the flux is correct. (Your parentheses look a little out of place.)
 

Related to Faraday's law with Calc 3 integration help

What is Faraday's law?

Faraday's law, also known as Faraday's electromagnetic induction law, states that a changing magnetic field in a closed loop of wire will induce a voltage in the wire.

How does Faraday's law relate to Calc 3 integration?

Calc 3 integration is used to calculate the total induced voltage in a closed loop of wire due to a changing magnetic field. This is done by integrating the magnetic field over the surface enclosed by the loop.

What are the units of measurement for Faraday's law?

The units of measurement for Faraday's law are volts (V) for the induced voltage, teslas (T) for the magnetic field, and square meters (m^2) for the surface area enclosed by the loop.

What is the significance of Faraday's law in science and technology?

Faraday's law is a fundamental principle in electromagnetism and is essential for understanding the functioning of many devices, such as generators, motors, and transformers. It also has practical applications in technologies such as wireless charging and electromagnetic induction heating.

Are there any real-life examples of Faraday's law in action?

Yes, there are many real-life examples of Faraday's law in action. One example is the use of induction cooktops, where a changing magnetic field creates an induced current in the cooking vessel, generating heat to cook food. Another example is the use of generators to convert mechanical energy into electrical energy using Faraday's law.

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