induced emf

Question

Loop A of radius r moves towards loop B (radius R >>r ) with a constant velocity in such a way that their planes parrallel . What is the distance between two loops in terms or R when the induced emf in loop A is maximum 

Answer 1

Hi Samagra!

 

So, we are looking at the emf induced in Loop A by Loop B -- which must have some current, assumed constant, running in it.  If it were not constant, we would need to know about its dependence on time, and we are not told anything about that, so I will proceed on the assumption that it is constant.

 

The consequences of making the radius of Loop A much, much smaller than the radius of Loop B are twofold:

 

1.  Loop A only receives flux from the axial magnetic field of Loop B, for which there is a derived formula; and,

2.  Loop A's induced emf, and resulting induced current and magnetic field in Loop A, do not induce a significant countercurrent in Loop B, which would affect how the loops interacted at later times

 

The induced emf in Loop A, ε_A, follows Faraday's law:

 

ε_A = N(dΦ_A/dt)

 

where N = number of loops = 1 in this case

Φ_A = magnetic flux through Loop A = B_BA_Acos(θ)  with B_B = magnetic field due to Loop B; A_A = area of Loop A, and θ = angle between Loop B's magnetic field and Loop A's area vector.

 

Now, if we consider ONLY the axial field from Loop B (because Loop A is so small, that is all it is seeing), then the orientation between the magnetic field and the area of Loop A will not change.  This means cos(θ) is constant in the flux expression.  Also, the area of Loop A is not changing, so A_A is constant.

 

Hence, the only quantity that is changing with time, as Loop A is moved toward Loop B, is B_B.  Therefore, (dΦ_A/dt) basically reduces to some constants times (dB_B/dt).  So let's calculate (dB_B/dt) first.

 

B_B(z) = C(1/(z²+R²)^3/2)  (C = a constant)

 

where z = Loop A's distance from Loop B (i.e. the distance between the loops at a given instant), and R is Loop B's radius, as given in the problem.

 

(you can look up, in the derived expression for a current loop's axial magnetic field, what C is, but it will not ultimately matter to the solution of this problem)

 

We can figure out (dB_B/dt) from the above expression using the differentiation rule for powers, and the chain rule:

 

(dB_B/dt) = C((-3/2)(z²+R²)^-5/2(2z)(dz/dt))

 

[note:  dz/dt is just the constant velocity -v (negative because z is decreasing in time as the Loop A approaches Loop B) with which Loop A approaches Loop B]

 

Simplifying a bit:

 

(dB_B/dt) = C(-3)(z²+R²)-^5/2(z)(-v)

 

The -3 and -v are both constant, and so can be absorbed into that front constant, which we may now call C'

 

(dB_B/dt) = C' (z/(z²+R²)^5/2)

 

So the value of ε_A depends on this quantity (and other constants).  To determine when ε_A is maximum, we can differentiate IT with respect to time, and then set the derivative equal to zero.  We can then solve for a value of z that makes this true.

 

Because ε_A depends on (dB_B/dt), (dε_A/dt) depends on (d/dt)(dB_B/dt).  So we need to differentiate (dB_B/dt) once more with respect to time.

 

(d/dt)(dB_B/dt) = (d²B_B/dt²) = d/dt(C'(z/(z²+R²)^5/2))

 

Using the power, quotient, and chain rules for differentiation, this becomes:

 

(d²B_B/dt²) = C'((z²+R²)^5/2(dz/dt) - z(5/2)(z²+R²)^3/2(2z)(dz/dt))/(z²+R²)⁵

 

The dz/dt terms in the numerator are again constants equal to -v, and can be factored out into that front constant, which we can now call C''.  Simplifying what is left inside the parentheses, we get:

 

(d²B_B/dt²) = C''((z²+R²)^5/2 - 5z²(z²+R²)^3/2)/(z²+R²)⁵

 

The reason we did not particularly care about the exact expression for those constants now comes into play.  All we care about is setting (dεA/dt) = 0 and solving for a value of z that makes this true.  This means, solving for a value of z that makes (d²B_B/dt²) = 0.  The only way this happens is if the numerator in the previous expression is equal to 0.  So, we set that numerator equal to zero, and solve for z:

 

(z²+R²)^5/2 - 5z²(z²+R²)^3/2  = 0

 

You can then solve this for z in terms of R.  I obtained z = (+ or -) R/2.

 

[NOTE: we would expect this kind of symmetry in the solution, since the magnetic field of Loop B is symmetric along the axis of the loop]

 

Mathematically, to assure this is a maximum, we would have to take the time derivative of (d²B_B/dt²) and show that it is negative when z = the value you solved for above.  But I am confident enough in the physics of this system (and busy enough) that I will leave that for you to confirm.

 

Hope this gets you on your way!  Just let me know if you have any questions about the myriad details.