circuit analysiselectromagnetismsolenoid
The current in LR circuit increases like this, I could understand the math which produces the graph by making sure the voltage sum is zero. But conceptually, I don't know what actually happens in circuit. Since inductance is caused by change in current, I don't understand why the current would go up by itself and I don't understand why the induced current continue to decrease.
I agree that emf is 1.5V but I didn't like the formula used. It should be: -
emf = B.A/t (flux density x area divided by time)
Also you have used 0.01m as the length of one side and this is incorrect - it is 0.1m. All the same you got the correct answer of 1.5 volts!!
Then you go and say the induced current is 0.015 amps - this is incorrect it is 0.15 amps i.e. 1.5 volts divided by 10 ohms.
Direction of current induced can be gleaned from this: -
The image above is for a loop getting bigger and this is the same as the field increasing linearly whilst the loop stays the same size. Link. If the loop were getting smaller, this would be equivalent to the scenario in the question and therefore the direction of current would be opposite to that shown in the diagram above.
So the induced current is subtracting from the standing current of 0.5 amps to produce a current of 0.35 amps. Or just look at it in terms of the net voltage being 3.5 volts across a ten ohm resistor.
Personally, I have never understood the "induced-emf" line of thinking. To me, it is easier, and much easier not to get it wrong, from the circuit and equations point of view.
I am simplifying the solenoid to be just an inductor. The equation for Circuit 1 is: $$ V_L = V_1 = L \frac{di}{dt} $$ It tells us right away that the voltage across L must be V1 and never goes to zero. The positive change of current never stops and i goes to infinity. Obviously, this is a bad model.
The obvious improvement to the model (unless the battery is underpowered) is to include the
ohmic resistance that you want to neglect. That is Circuit 2. The equation:
$$ V_1 = V_L + V_R = L \frac{di}{dt} + R i $$
Initially, i is zero. All of V1 is across L. i increases.
As i increases, voltage drops across R. Voltage across L decreases, rate of i increase slows down.
Eventually, when i reaches the level where R x i = V1. Voltage across L becomes 0.
If you solve for \$V_L\$ and \$i\$, you get something like: $$ V_L = V_1e^{-\frac{R}{L}t} $$ $$ i = \frac{V_1}{R} (1 - e^{-\frac{R}{L}t}) $$ \$V_L\$ is not the voltage across the solenoid, we need to include the voltage across R for that, the total would then be \$V_1\$.
Best Answer
Shortly:
Voltage V drives the current. Induction generates an opposite voltage in the coil (actually in the magnetic field around the coil) and that prevents the current to step infinitely fast to the final limit value V/R which is stated by Ohm's law. As the current grows the net voltage over the inductor diminishes because the resistor takes off amount IR. Thus the growth gradually decays.
More:
Induction is one of the basic physical facts between electric and magnetic fields. It has no proven mechanical explanation, it's a fact that can be measured and seems to obey certain vector field differential equations.
In mechanics we also have some as much unexplained facts, for ex. the gravity. It can be measured and it obeys certain differential equations but it's impossible to see what's the basic reason why mass causes gravity.
Those field equations of the induction can be applied to different structures and that produces limited special cases. In coils and wires (=inductors) the induction occurs as a relation between the voltage, current, time and one geometry and material dependent constant (=the inductance). The basic differential equation for induction in inductors is:
"The voltage between the terminals = the inductance (Henries) multiplied by the growth rate of the current (Amperes per a second)" or U = L(dI/dt)
See this old case for more practical explanations: How does the inductor ''really'' induce voltage?
Non-ideal inductors have some resistance and also capacitance, but they can be modeled as separate series and parallel parts if one calculates how an inductor behaves in a circuit.
All possible circuits in all cases have such voltages and currents that every circuit law is satisfied in every moment. Generally the voltages and currents vary in time because many circuit elements, for ex. inductors have operation law where one quantity causes another quantity to grow or diminish. The currents and voltages vary in a way which depends on the initial conditions and what variations possible signal sources force to happen.
In your circuit Kirchoff's law states: V= IR + L(dI/dt) Solving that differential equation gives your exponential equations where the current gradually grows towards the limit caused by ohm's law. Without the resistor the current would grow infinitely with rate (dI/dt)=V/L