The Laplace force

What Nour says

1. 1. The Laplace force: definition and direction

   Today, the whole chapter: "The Laplace force", Physics, Year 3, Mathematics section.

   We will cover the force itself, its formula, the DC motor, the Barlow wheel, and the energy it gives.

   Let's start with the Laplace force.

   A wire carries an electric current, and it sits inside a magnetic field.

   The magnetic field pushes on that current-carrying wire with a force.

   This push is called the Laplace force, that is the force a magnetic field exerts on a current-carrying wire.

   We already know about magnetic interaction from an earlier chapter, and this force is one example of it.

   To see this force, we need three things together: a current, a magnetic field, and a wire placed inside it.

   Let's draw the full picture on the board now.

   Here is a straight wire of length L, carrying a current of intensity I.

   Around it, a uniform magnetic field B points in this direction.

   The Laplace force F appears, perpendicular to both the wire and the field.

   This perpendicular direction is the key idea behind this force.

   Here is why the direction works this way.

   The force is always perpendicular to the plane that contains the wire and the field.

   To find its exact sense, we use the right-hand rule, like for other magnetic effects.

   Point your fingers along the current, curl them toward the field, your thumb gives the force.

   If the wire lies along the field itself, the angle between them is zero.

   In that special case, there is no Laplace force at all.

   The force only appears when the wire crosses the field lines, not when it runs along them.

   Now let's work through a full example, like in a bac exam problem.

   A straight horizontal wire of length 20 centimeters carries a current of 3 amperes.

   It sits inside a uniform magnetic field of 0.5 tesla, perpendicular to the wire.

   We want to find the direction of the Laplace force on this wire.

   First, draw the wire carrying current I from left to right.

   Then draw the magnetic field B pointing straight up, perpendicular to the wire.

   Since the angle between the wire and the field is 90 degrees, the force reaches its maximum size.

   Using the right-hand rule, fingers along I, curling toward B, the thumb points out of the page.

   So the Laplace force on this wire points out of the page, toward us.

   We frame this result: the force is perpendicular to the plane of the wire and the field.

   Its size also depends on the angle between the current and the field.

   Let's state the key result clearly now.

   The Laplace force acts on a current-carrying wire placed in a magnetic field.

   Its direction is always perpendicular to both the current and the magnetic field.

   We find that direction using the right-hand rule, and it vanishes when the wire is parallel to the field.

   In what direction does the Laplace force always point, relative to the wire and the field?

2. 2. The formula F = BIL sin(alpha)

   Now let's give the Laplace force its exact formula.

   The size of the force depends on three things: the field, the current, and the wire's length.

   Here F is the force, in newtons.

   B is the magnetic field strength, in teslas.

   I is the current intensity, in amperes.

   L is the length of wire inside the field, in meters.

   And alpha is the angle between the wire's current and the magnetic field.

   Here is why the sine of the angle matters so much.

   When the wire is perpendicular to the field, alpha is 90 degrees, and sine equals one.

   That gives the strongest possible force for these values of B, I, and L.

   When the wire lies parallel to the field, alpha is zero, and sine equals zero.

   Then the force disappears completely, matching what we saw before.

   Let's solve a full bac-style problem with this formula now.

   A wire of length 50 centimeters carries a current of 4 amperes.

   It sits in a magnetic field of 0.2 tesla, making an angle of 30 degrees with it.

   We want to calculate the size of the Laplace force on this wire.

   We apply the formula F equals B times I times L times sine of alpha.

   We substitute the values: 0.2 tesla, 4 amperes, 0.50 meter, sine of 30 degrees.

   Sine of 30 degrees equals 0.5, so we multiply everything together.

   This gives a force of 0.2 newton.

   We frame this final answer, in newtons, as the formula always requires.

   Let's state the key result of this formula clearly.

   The Laplace force equals B times I times L times the sine of the angle.

   The force is strongest when the wire sits perpendicular to the field.

   And it vanishes when the wire runs parallel to the magnetic field lines.

   What is the force when the wire makes a 0-degree angle with the field?

3. 3. The DC electric motor

   Now let's move to a real machine that uses the Laplace force.

   It is called the DC electric motor, that is a device that turns electric current into rotation.

   It uses the Laplace force on a loop of wire to make that loop spin.

   Let's draw the basic setup of this motor.

   A rectangular loop of wire can rotate around a fixed axis through its center.

   A magnet creates a uniform magnetic field B across the loop's space.

   Current I flows through the loop, up one side and down the other.

   On each side, the Laplace force pushes, but in opposite directions.

   These two opposite forces create a turning effect, a torque, on the loop.

   Here is why the loop keeps spinning in one direction.

   The two forces are equal, opposite, and not on the same line, so they form a couple.

   This couple of forces rotates the loop around its axis.

   A part called the commutator reverses the current every half turn, keeping the push in the same sense.

   Without this reversal, the loop would stop at the halfway point instead of continuing.

   Now let's apply this to a worked bac-style question.

   A rectangular loop has two sides of length 8 centimeters, each carrying current 2 amperes.

   The loop sits in a magnetic field of 0.3 tesla, perpendicular to the current in each side.

   We want the Laplace force on just one side of the loop.

   We use the same formula: F equals B times I times L times sine of alpha.

   We substitute: 0.3 tesla, 2 amperes, 0.08 meter, and sine of 90 degrees equals one.

   Multiplying these gives a force of 0.048 newton on that side.

   The opposite side feels the same size of force, but pointing the other way.

   Together, these two forces make the loop turn around its axis.

   Let's state the key result about this motor now.

   The DC electric motor turns current into rotation using the Laplace force on a loop.

   Two opposite forces on the loop's sides form a couple that spins it.

   The commutator reverses the current each half turn, so the loop never stops turning.

   Why does the motor need the commutator to reverse the current?

4. 4. The Barlow wheel

   Now let's see another device that spins thanks to the Laplace force.

   It is called the Barlow wheel, that is a metal wheel with spokes that spins in a magnetic field.

   A current flows from the center out to the edge through one spoke, dipped in mercury.

   Let's draw this wheel and see how it turns.

   Here is the wheel, a circle that can rotate freely around its center.

   One spoke carries the current I, from the center out to the rim.

   A magnetic field B is set perpendicular to the wheel's plane.

   The Laplace force acts on that current-carrying spoke, pushing it sideways.

   This force is tangent to the wheel, so it makes the wheel spin around its center.

   Here is why the wheel keeps spinning in this setup.

   As the wheel turns, a new spoke always sits where the current flows, repeating the push.

   So the Laplace force keeps acting, and the wheel rotates continuously, not just once.

   Let's solve a bac-style problem about this wheel now.

   The Barlow wheel has a radius of 10 centimeters and carries a current of 5 amperes.

   It sits in a magnetic field of 0.4 tesla, perpendicular to the spoke.

   We want the Laplace force acting on the spoke at the rim.

   We use the formula again: F equals B times I times L times sine of alpha.

   Here L is the radius of the wheel, since the spoke runs from center to rim.

   Multiplying these values gives a force of 0.2 newton.

   This force acts tangent to the rim, turning the wheel around its axis.

   Let's state the key result about this wheel clearly.

   The Barlow wheel spins because the Laplace force pushes each spoke sideways as current flows.

   The force stays tangent to the wheel, so the rotation never stops on its own.

   Why does the force on the spoke keep the wheel turning without stopping?

5. 5. Energy and power of the Laplace force

   Now let's look at the energy and power of the Laplace force.

   When this force moves a wire, it does work, that is it transfers energy to the wire.

   Here W is the work done, in joules.

   F is the Laplace force, in newtons, and d is the distance the wire moves, in meters.

   Power is how fast this work is done, measured in watts.

   Here is why this energy matters in a motor.

   The electric current brings energy in, and the Laplace force turns part of it into movement.

   This is how a motor's electric power becomes mechanical power, driving the rotation.

   Some energy is lost as heat in the wire, so not all of it becomes motion.

   Let's work through a bac-style problem on this energy now.

   A wire feels a Laplace force of 0.3 newton and moves 0.4 meter along the push.

   This motion takes a time of 2 seconds.

   We want the work done by the force, and the power it delivers.

   We use the formula: work equals force times distance.

   We substitute: 0.3 newton times 0.4 meter, giving 0.12 joule.

   Now for power, we divide work by time: 0.12 joule over 2 seconds.

   This gives a power of 0.06 watt delivered by the Laplace force.

   We frame both results: the work in joules, and the power in watts.

   Let's state the key result about energy and power now.

   The Laplace force does work when it moves a wire, and that work is force times distance.

   Power tells us how fast this work happens, and it equals work divided by time.

   This energy transfer is what drives motors and wheels in real machines.

   What is the power if the same work is done in half the time?

   That completes the whole chapter on the Laplace force.

   We saw its definition, its formula, the motor, the Barlow wheel, and its energy.

   Keep practicing the formula with units, and the direction rule, for your bac exam.