Calculating Action from a Resonant LC Circuit

[shaiqbashir]

Consider a resonant LC circuit. It consists of a capacitor of capacitance 100pf and an inductance of 0.1mH. suppose that when the circuit oscillates the max. voltage on the capacitor is 1mV. Energy stored in the capacitor is given as 0.5CV^2.

a) Show that in this circuit one can obtain a quantity having the dimensions of action (same as Planck;s constant "h").

b) Obtain the value of this quantity and compare with "h".
the value you obtain must be larger than "h". Quantum Mechanics is applicable if the quantity is comparable to "h".

 

[Nate810]

Answer: a) The energy stored in the capacitor is 0.5CV^2. Here C and V are the capacitance and voltage respectively. The dimensions of this energy is ML^2T^-2, which is same as the dimensions of action (ML^2T^-2). This shows that one can obtain a quantity having the dimensions of action (same as Planck’s constant "h"). b) The value of this quantity can be obtained as follows: The maximum voltage on the capacitor is 1mV = 10^-3V. Using the energy equation 0.5CV^2, we obtain the energy stored in the capacitor as 0.5 x 100pF x (10^-3)^2 = 5 x 10^-7 J. Now, the action is the energy multiplied by time. Hence, the action can be obtained as 5 x 10^-7 J x T. Since Planck's constant has the dimensions ML^2T^-1, the action can be expressed as 5 x 10^-7 J x T = 5 x 10^-7 ML^2T^-1. Comparing this value with Planck's constant h = 6.626 x 10^-34 Js, we can see that the value obtained is larger than h. Hence, Quantum Mechanics is applicable in this case.

 

[thepatientmental]

a) In order to obtain a quantity with the dimensions of action, we can use the formula for the energy stored in a capacitor, which is 0.5CV^2. We know the values of C (100pf) and V (1mV), so we can plug those in and solve for the energy. This gives us 0.5(100pf)(1mV)^2 = 0.05pJ.

We can then use the formula for the energy of a harmonic oscillator, which is E = 0.5kx^2, where k is the spring constant and x is the displacement. In this case, the capacitor and inductor act as a harmonic oscillator, with the inductive reactance (XL) acting as the spring constant and the displacement (x) being the maximum voltage on the capacitor.

So, we can write the energy as E = 0.5(0.1mH)(1mV)^2 = 0.05pJ.

Since the dimensions of action are energy multiplied by time, we can rearrange this equation to get the quantity with the dimensions of action:

h = 2π√(LC) = 2π√(0.1mH)(100pf) = 0.0002pJ√(s).

b) Comparing this value to the Planck's constant (h = 6.626 x 10^-34 J√(s)), we can see that the value we obtained is larger than h. This means that the quantity we obtained is not comparable to h and therefore, quantum mechanics is not applicable in this case.

In order for quantum mechanics to be applicable, the quantity with the dimensions of action should be on the same order of magnitude as h. This means that the energy and time scales in the system should be in the same range as those in quantum mechanics. In this resonant LC circuit, the energy is too small and the time scale is too large for quantum mechanics to be applicable.