# Current in an RL-Circuit

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• Fernando Rios
However this will not be the case in the real world where there is always self inductance in RC circuits.f

#### Fernando Rios

Why is initial current zero in an RL ciruit with an emf, but it is not in a charging RC circuit?

Delta2
It starts from zero in an RC circuit as well, but in a very small time ##\Delta t## (this time depends on the self inductance of the circuit which is small, and on the dimensions of the circuit/speed of light-practically this time ##\Delta t## is zero, theoretically is not zero) it climbs up to a value ##\frac{E}{R}## and then it starts exponential decay according to what is well known for an RC circuit. Most books omit this phase of the time ##\Delta t## because it is very small, practically zero as I said.

It starts from zero in an RC circuit as well, but in a very small time ##\Delta t## (this time depends on the self inductance of the circuit which is small, and on the dimensions of the circuit/speed of light-practically this time ##\Delta t## is zero, theoretically is not zero) it climbs up to a value ##\frac{E}{R}## and then it starts exponential decay according to what is well known for an RC circuit. Most books omit this phase of the time ##\Delta t## because it is very small, practically zero as I said.

Delta2
In case you are interested for an explanation that uses more math take a look at the following wikipedia link
https://en.wikipedia.org/wiki/RLC_circuit#Series_circuit

Any RC circuit is actually an RLC circuit where L is the self inductance of the RC circuit (there is always self inductance to a circuit) which is small though.

So the damping factor ##\zeta=\frac{R}{2}\sqrt\frac{C}{L}## becomes large (because L is small and in the denominator) so it usually is ##\zeta>>1## for a typical RC circuit (which is actually an RLC circuit as i said before). So the circuit response is the overdamped response, that is exponential decay without going into oscillations.

Make sure to check the diagram with the different curves of current ##I(t)## for different values of ##\zeta## in that section of wikipedia link. Notice how all curves start from ##I(0)=0## and how current rises fast to a maximum value before starting the exponential decay.

You might like to think about a mechanical analogy. For instance, a spring (with zero mass) is like a capacitor; if we apply a force, it changes length instantly and stores the supplied energy as elastic PE. On the other hand, a mass is like an inductor; when we apply a force, it accelerate slowly (F=MA) and stores the applied energy as kinetic energy.
The mechanical laws are an analogue of the electrical ones. For instance, for an inductor Energy =LI^2/2 whereas for a mass Energy=Mv^2/2.

Delta2
You might like to think about a mechanical analogy. For instance, a spring (with zero mass) is like a capacitor; if we apply a force, it changes length instantly and stores the supplied energy as elastic PE. On the other hand, a mass is like an inductor; when we apply a force, it accelerate slowly (F=MA) and stores the applied energy as kinetic energy.
The mechanical laws are an analogue of the electrical ones. For instance, for an inductor Energy =LI^2/2 whereas for a mass Energy=Mv^2/2.
What you saying is almost correct, It is just that even if we attach no mass to a spring, the mass of the spring itself will prevent it of changing length instantly. However if we also assume that the spring is massless, then there is no way to prevent the mini apparent paradox that the spring will change length instantly.

In the RC circuit case if we assume that the circuit has no self inductance and also has zero dimensions (lumped model) then there is nothing from preventing the mini paradox that the current will have initial value E/R at time t=0.

sophiecentaur