Non-Conserative Induced Electric Field

We always encounter situations that deviate from the normal. Doesn’t it seem bizarre that a voltmeter connected across two points can actually show two different values depending on how it is connected? Sounds strikingly similar to the mechanical equivalent, where friction does varying amounts of work between two points depending on the path taken to move a body. Doesn’t it? Yes. The culprit is a non-conservative force in action.

Consider a circuit with a 10V battery and two resistors R₁=80 ohms and R₂ =20 ohms. If we are to calculate the current i through the circuit, applying the Kirchhoff’s loop law yields this:

If we were to find the potential difference between A and B as shown,we have two paths at our disposal, namely, 1 & 2:

By following path 1, we get V_a - V_b = 80(0.1) = 8 V

By following path 2,we get V_a - V_b = 10 - 20(0.1) = 8 V

Any path taken gives the same value of V_a - V_b. With further deductions, we can understand that this must hold good because electric field produced by the battery or, in general, any static charge is conservative.

Now, consider the case of magnetically induced electric fields. Are they conservative or non-conservative? To analyze their nature, utilize a similar setup without the battery. Instead, there is a changing magnetic field through the loop.

Consider the instant when the rate of change of the flux through the loop is exactly 10V. According to Faraday’s Law, EMF induced in the loop will be 10V. Again, we find the current i through the loop, which comes out to be 0.1 A. Let the two digital voltmeters be connected across the two resistors as shown.

We may judge that both the voltmeters will show the same reading as they are connected across the same points. Surprisingly, this is not so!

The voltmeters show entirely different readings .This is rationalized by these calculations.

We found that the current in the circuit is 0.1A. So, if we go trace path 1 we get

V_a - V_b = 80(0.1) = 8 V,

which will be the same reading as that of the voltmeter V1.

By path 2,we get

V_a - V_b = -20(0.1) = -2 V

Evidently, this dissimilarity points to the fact that we ‘removed’ the battery voltage source (and replaced it with something else). So, voltmeter V₂ shows -2V.

Summing it up, we get two voltmeters attached across the same two points showing different readings. What is the mystery behind this baffling complication?

Well, magnetically induced electric fields are non-conservative. When we calculate potential difference between two points taking two different paths, we get different answers unlike the electric field created by charges. Thus, magnetically induced fields are non-conservative in nature and forms loops. This property is different from the conventional electric fields, which emerge at a positive charge and end at a negative charge. Can you find out some more anomalies of non-conservative fields?