Solution Found!
If the resistance in the RL circuit of Figure 3.13(a) is
Chapter 3, Problem 4E(choose chapter or problem)
If the resistance in the 𝑅𝐿 circuit of Figure 3.13(a) is zero, show that the current 𝐼(𝑡) is directly proportional to the integral of the applied voltage 𝐸(𝑡). Similarly show that if the resistance in the 𝑅𝐶 circuit of Figure 3.13(b) is zero, the current is directly proportional to the derivative of the applied voltage. (In engineering applications, it is often necessary to generate a voltage, rather than a current, which is the integral or derivative of another voltage. Group Project E shows how this is accomplished using an operational amplifier.)
Questions & Answers
QUESTION:
If the resistance in the 𝑅𝐿 circuit of Figure 3.13(a) is zero, show that the current 𝐼(𝑡) is directly proportional to the integral of the applied voltage 𝐸(𝑡). Similarly show that if the resistance in the 𝑅𝐶 circuit of Figure 3.13(b) is zero, the current is directly proportional to the derivative of the applied voltage. (In engineering applications, it is often necessary to generate a voltage, rather than a current, which is the integral or derivative of another voltage. Group Project E shows how this is accomplished using an operational amplifier.)
ANSWER:
Solution :
Step 1 :
- :
In this problem we have to show that the current I(t) is directly proportional to the integral of the applied voltage.
We use the below laws of physics in the figure 3.13
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According to Ohm’s law, the voltage drop
across a resistor is proportional to the current I passing through the resistor.
.’. …………..(1)
The proportionality constant R is called the resistance.
-
According to Faraday’s law Lenz’s law that the voltage drop
across an inductor is proportional to the instantaneous rate of change of the current I.
.’.
……………(2)
The proportionality constant L is called the inductance.