The inductor; the RL circuit

What Nour says

1. 1. The inductor and electromagnetic induction

   Today, the whole chapter: "The inductor; the RL circuit", Physics, Year 4, Mathematics section.

   We will build this step by step: what a coil does, the law that fixes its direction, then the full RL circuit and its stored energy.

   Each part has a drawing, a worked example, and a short question to check your understanding.

   Let's start with the coil, a component made of a wire wound many times around a tube.

   We draw it as a spiral of wire, with two ends to connect it to a circuit.

   This component is called an inductor, and in a circuit it is symbolized by the letter L.

   Its key property shows up only when the current through it changes over time.

   This property is called electromagnetic induction, which means a changing current creates an opposing effect inside the coil.

   You already know the capacitor, from the RC circuit chapter, which stores charge on its plates.

   The coil is different: it reacts against changes in current, not against charge.

   Think of the coil as a component that resists sudden change, almost like inertia for electric current.

   If the current tries to jump suddenly, the coil fights back and slows that change down.

   This is why a coil never lets current change instantly, even if the voltage source switches on abruptly.

   Let's see this with a worked example, close to what you will see on the bac exam.

   A coil of inductance L equals 0{,}5 henry carries a current that rises from 0 ampere to 2 amperes in 0{,}1 second.

   We want the average rate of change of the current, written delta i over delta t.

   We divide: 2 amperes divided by 0{,}1 second.

   This gives 20 amperes per second, a fast change in current.

   Because this rate is large, the coil reacts strongly, producing a noticeable induced effect inside it.

   In the next sections, we will see exactly how to describe and calculate that induced effect.

   For now, remember this: a coil is a wire loop that reacts whenever the current through it changes.

   The faster the current changes, the stronger this reaction from the coil becomes.

   This whole phenomenon, the reaction to changing current, is what we call electromagnetic induction.

   What property of the coil appears only when the current changes over time?

2. 2. Lenz's law and the direction of induced effects

   Now let's move to Lenz's law, which tells us the direction of the induced effect in a coil.

   That is, this law says the coil always acts to oppose the change that causes it.

   If current tries to increase, the coil pushes back against that increase.

   If current tries to decrease, the coil pushes to keep it from falling.

   Let's draw a simple circuit to picture this: a coil connected to a switch and a battery.

   At the moment we close the switch, the current wants to jump from 0 ampere instantly.

   The coil reacts right away, opposing this sudden rise, and slows the current's growth.

   This is why the current rises smoothly after closing the switch, never with a sharp jump.

   Here is a limit case: without any coil, the current would jump instantly to its final value.

   With the coil present, that jump is smoothed into a gradual rise over time.

   Now let's apply this idea to a worked example, similar to a bac-style question.

   In a circuit, the switch closes at a time we call t equals 0 second.

   Just before closing, the current in the coil was 0 ampere, since the circuit was open.

   The question asks: what is the current in the coil just after closing, at t equals 0 plus?

   Since current through a coil is always continuous, it cannot jump at the switching instant.

   So the current right after closing equals the current right before: both are 0 ampere.

   This continuity rule is a direct consequence of Lenz's law opposing any sudden change.

   Remember this key result: the induced effect in a coil always opposes the change that creates it.

   This opposition is what keeps the current continuous, never jumping, through any coil.

   We use this rule constantly when we study how an RL circuit responds to switching.

   Can the current through a coil change instantly at the moment a switch closes?

3. 3. The induced electromotive force of a coil

   Now we study the induced electromotive force of a coil, the voltage it creates when current changes.

   That is, this is the extra voltage a coil produces whenever the current through it changes over time.

   We write this induced voltage as e, measured in volts, written V.

   The formula links e to the inductance L and to how fast the current changes.

   Here, L is the inductance of the coil, measured in henry, written H.

   And di over dt is the rate of change of current, in amperes per second.

   The minus sign in the formula is exactly Lenz's law written in mathematical form.

   It shows the induced voltage always works against the change in current, not with it.

   When the current is steady, not changing at all, di over dt equals 0 ampere per second.

   So a coil only produces this voltage during a change, never during a steady current.

   Let's apply this formula in a worked example like one from the bac exam.

   A coil has inductance L equals 0{,}2 henry.

   Its current changes at a constant rate of 5 amperes per second.

   We need to find the induced electromotive force e, in volts.

   We apply the formula: e equals minus L times di over dt.

   We substitute: minus 0{,}2 henry times 5 amperes per second.

   This gives e equals minus 1 volt.

   The negative sign tells us the induced voltage opposes this particular rise in current.

   On the bac exam, you often just need the magnitude, so here it is 1 volt.

   Remember this key formula: the induced electromotive force equals minus L times di over dt.

   It depends only on the inductance and on how fast the current changes, never on the current's value itself.

   This formula is the heart of how a coil behaves in any circuit we study next.

   What is the induced e.m.f.

   when the current in a coil stays perfectly constant?

4. 4. Response of an RL circuit to a voltage step

   Now we study the response of an RL circuit to a voltage step, that is, how current builds up when we switch on a steady voltage.

   Let's draw this circuit: a battery, a resistor R, and a coil L, all in series with a switch.

   At time t equals 0 second, we close the switch and apply a constant voltage E.

   We write the voltage law around the loop: E equals the resistor's voltage plus the coil's voltage.

   This equation describes exactly how the current i builds up over time, second by second.

   Solving it gives us the current as a function of time, rising toward a final value.

   Here, tau is the time constant, which we will study in detail in the next part.

   At the start, t equals 0 second, the current is still 0 ampere, by continuity.

   After a long time, the current reaches its final steady value, E over R.

   Let's see a graph of this rising current, starting at 0 ampere and climbing smoothly.

   The curve rises quickly at first, then flattens out as it nears the final current.

   The dashed line marks the final value, E over R, which the curve approaches but never quite reaches.

   Now let's work through a bac-style example on this exact circuit.

   An RL circuit has a battery of E equals 6 volts and a resistor R equals 30 ohms.

   We want the final current, once the current has stopped changing, reached a steady state.

   We use the formula: final current equals E over R.

   We substitute: 6 volts divided by 30 ohms.

   This gives a final current of 0{,}2 ampere.

   Notice this final value does not depend on the coil's inductance L at all.

   The inductance only controls how fast the current gets there, not where it ends up.

   Remember this key result: current in an RL circuit rises smoothly from 0 ampere to E over R.

   This smooth, gradual rise is the signature response of a coil to a sudden voltage step.

   In the next part, we study tau, the number that tells us how fast this rise happens.

   What current flows through the coil right at the instant the switch closes?

5. 5. The time constant tau

   Now let's study tau, the time constant, which tells us how fast the current rises in an RL circuit.

   That is, tau measures the typical time the current needs to approach its final value.

   The formula for tau is simple: inductance L divided by resistance R.

   Tau is measured in seconds, written s, just like any other time.

   A bigger inductance L makes tau larger, so the current rises more slowly.

   A bigger resistance R makes tau smaller, so the current rises more quickly.

   Let's draw the current curve again and mark tau directly on it.

   The curve rises from 0 ampere toward its final value, E over R, following the same shape.

   At the special time t equals tau, the current reaches about 63 percent of its final value.

   This single point is the practical way to read tau straight off an experimental graph.

   In practice, after about 5 times tau, the current is so close to final value we call it steady.

   Let's work through a bac-style example to find tau from given circuit values.

   A coil has inductance L equals 0{,}4 henry, in a circuit with resistance R equals 20 ohms.

   We calculate tau using the formula: L divided by R.

   This division gives tau equals 0{,}02 second.

   This means the current reaches 63 percent of its final value after just 0{,}02 second.

   Remember this key formula: tau equals L over R, and it sets the speed of the whole rise.

   A graph's steepness and the circuit's L and R values are directly linked through this one number.

   If we double the resistance R, what happens to the time constant tau?

6. 6. Energy stored in an inductor

   Now the last section: the energy stored in an inductor, the energy a coil keeps while current flows through it.

   That is, this energy is stored in the magnetic field the coil creates around itself.

   We write this energy as E sub L, measured in joules, written J.

   The formula is: E sub L equals one half times L times i squared.

   Here L is the inductance in henry, and i is the current in amperes, at that instant.

   This looks a lot like the capacitor's stored energy formula from the RC circuit chapter.

   But here, energy depends on current, not on voltage, since the coil's field comes from current.

   When current is 0 ampere, the coil stores 0 joule, since there is no magnetic field at all.

   As current grows larger, the stored energy grows much faster, since it depends on i squared.

   This energy builds up while the current rises, and it is released if the current later falls.

   Let's work through a bac-style example to calculate this stored energy.

   A coil has inductance L equals 0{,}1 henry and carries a steady current of 3 amperes.

   We want the energy stored in this coil, E sub L, in joules.

   We apply the formula: one half times L times i squared.

   We square the current first: 3 amperes squared equals 9.

   Now we multiply: one half times 0{,}1 times 9.

   This gives a stored energy of 0{,}45 joule.

   This energy stays stored in the magnetic field as long as the current stays at 3 amperes.

   Remember this key formula: energy stored equals one half times inductance times current squared.

   This energy comes from the magnetic field, and it depends only on the current at that instant.

   This closes our tour of the coil, from induction, through Lenz's law, to the full RL circuit.

   If the current through a coil doubles, what happens to its stored energy?

   That completes our chapter on the inductor and the RL circuit.

   We moved from a single coil's reaction to current, through Lenz's law, to the full RL response and its stored energy.

   Practice the two exercises, always tracking units, and you will be ready for this topic on the bac.