RL Circuit Analysis: Comparing Resistor and Coil Voltages

[member 392791]

Homework Statement

At the moment t = 0, a 24.0-V battery is connected
to a 5.00-mH coil and a 6.00-V resistor. (a) Immediately
thereafter, how does the potential difference across the
resistor compare to the emf across the coil? (b) Answer
the same question about the circuit several seconds later.
(c) Is there an instant at which these two voltages are equal
in magnitude? If so, when? Is there more than one such
instant? (d) After a 4.00-A current is established in the
resistor and coil, the battery is suddenly replaced by a short
circuit. Answer parts (a), (b), and (c) again with reference
to this new circuit.

Homework Equations

The Attempt at a Solution

a) V_r = 0 at t=0
b) V_r = ε at t --> ∞

V_l = V_r at t=??

I tried doing -L di/dt = IR to solve for t and it got me nowhere

d) when the battery is removed,
I got V_r = ε at t=0 and V_r = 0 at t-->∞
, and still don't know if there is a time that V_r = V_l

 

[NascentOxygen]

The elements are in series, presumably.

I tried doing -L di/dt = IR to solve for t and it got me nowhere

If you can't solve a first-order differential equation, then steal someone else's solution from a textbook. The response of a first order system (whether R+L or R+C ) is something you need to practise until you can sketch the graph in your sleep.

 

[member 392791]

I know how to do the differential equation

the problem is that everything canceled out and got me back to R = R.

 

[NascentOxygen]

What is the general solution to the DE with L, R and V?

 

[Nickie]

I would approach this problem by first analyzing the circuit using Kirchhoff's laws and Ohm's law to determine the behavior of the circuit at different time points.

(a) Immediately after the battery is connected, the potential difference across the resistor will be equal to the emf across the coil. This is because the current in the circuit is initially zero and therefore there is no voltage drop across the resistor.

(b) After several seconds, the potential difference across the resistor will decrease as the current in the circuit increases. However, the emf across the coil will remain constant. Therefore, the potential difference across the resistor will be less than the emf across the coil.

(c) There will be an instant when the potential difference across the resistor will be equal to the emf across the coil. This will occur when the current in the circuit has reached its steady state value, which can be calculated using Ohm's law and Kirchhoff's laws. There may be multiple instants when this occurs, depending on the specific values of the circuit components.

(d) When the battery is replaced by a short circuit, the potential difference across the resistor will be zero, as there is no longer a voltage drop across it. The emf across the coil will also be zero, as there is no longer a potential difference across it. Therefore, the potential difference across the resistor will be equal to the emf across the coil at all times in this new circuit.

In summary, the potential difference across the resistor and the emf across the coil will be equal at different times depending on the initial conditions and the behavior of the circuit. A thorough analysis using relevant equations and laws is necessary to determine these time points accurately.