Overload Infinity Transformer?

Electricity and Magnetism Level 5

Refer to the figure shown. An E E - V V battery and infinitely many solenoids of 1 ! 1! , 2 ! 2! , 3 ! 3! and so on. The number of loops is connected in parallel. Another circuit has each of the infinite number of corresponding identical solenoids having N N loops and internal resistance R R connected in parallel. This circuit is placed near but not in contact with the former circuit.

In terms of the variables given, determine the total current induced when the switch is closed.

Details and Assumptions :

  • V V is read as Volts, the unit of potential difference.

  • All elements are Ohmic.

E N R ( e − 1 ) \dfrac{E}{NR} (e-1) ∞ \infty N E R ( e − 1 ) \dfrac{NE}{R} (e-1) E R ( e − 1 ) \dfrac{E}{R} (e-1) E N R e \dfrac{E}{NR} e N E R e \dfrac{NE}{R} e N E R e NERe E R \dfrac{E}{R}

Louie Dy by Louie Dy , 24, Philippines

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1 solution

Louie Dy
May 4, 2016

One of the equations on transformers is the following:
V 1 N 1 = V 2 N 2 \dfrac{V_1}{N_1} = \dfrac{V_2}{N_2}

By substitution, the transformation for each of the primary n ! n! -loop to secondary N N -loop leads to:

E n ! = V n N \dfrac{E}{n!} = \dfrac{V_n}{N}

V n = N E n ! V_n = \dfrac{NE}{n!}

By Ohm's Law (since the solenoids are Ohmic),

I n = N E n ! R I_n = \dfrac{NE}{n!R}

Then getting the sum:

∑ n = 1 ∞ I n \displaystyle\sum_{n=1}^\infty I_n

∑ n = 1 ∞ N E n ! R \displaystyle\sum_{n=1}^\infty \dfrac{NE}{n!R}

N E R ∑ n = 1 ∞ 1 n ! \dfrac{NE}{R} \displaystyle\sum_{n=1}^\infty \dfrac{1}{n!}

By Maclaurin Series,

N E R ( e − 1 ) \dfrac{NE}{R} (e-1)

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