4.11. SUMMARY

The total force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E, respectively is called the Lorentz force. It is given by the expression:
F = q (v × B + E)
The magnetic force q (v × B) is normal to v and work done by it is zero.

The Lorentz force is the overall force acting on a charged particle when it moves through both electric and magnetic fields. This equation shows that the total force (F) is dependent on the charge (q) of the particle, its velocity (v), and the magnetic field (B) it is moving through. The term (v × B) is the magnetic force and is perpendicular to the direction of the particle's movement, indicating that it does no work. This means the speed of the particle remains constant while its direction may change.

Imagine a charged particle, like an electron, as a soccer ball kicked along a field. The electric field is like a player pushing the ball in one direction, while the magnetic field is like a wind blowing sideways. The wind changes the direction of the ball without increasing its speed.

A straight conductor of length l and carrying a steady current I experiences a force F in a uniform external magnetic field B,
F = I l × B
where |l| = l and the direction of l is given by the direction of the current.

When a straight wire, carrying an electric current and placed in a magnetic field, will experience a force. This force's magnitude depends on the current (I), the length of the wire (l), and the strength of the magnetic field (B). The cross-product (l × B) indicates that the force acts at an angle to both the direction of the current and the magnetic field, highlighting the importance of orientation in this scenario.

Think of the wire as a boat moving through water. The current carries the boat forward (like the current in the wire) while the magnetic field acts like a current in the water, pushing the boat sideways. The angle of the push (the force) varies depending on how the boat is aligned with the water flow.

In a uniform magnetic field B, a charge q executes a circular orbit in a plane normal to B. Its frequency of uniform circular motion is called the cyclotron frequency and is given by:
n = qB / (2πm)
This frequency is independent of the particle’s speed and radius. This fact is exploited in a machine, the cyclotron, which is used to accelerate charged particles.

When a charged particle moves in a uniform magnetic field, it moves in a circular path. The frequency of this circular motion, known as the cyclotron frequency, depends only on the charge of the particle (q), the strength of the magnetic field (B), and the mass of the particle (m). Interestingly, this frequency does not depend on how fast the particle is moving or the size of its path. This principle is utilized in devices called cyclotrons which accelerate particles for experiments or medical applications.

Imagine a child on a merry-go-round in a park. No matter how fast the child runs around the edge, the speed at which the merry-go-round turns (the 'frequency') remains constant, depended only on the 'push’ given to it (like qB).

The Biot-Savart law asserts that the magnetic field dB due to an element dl carrying a steady current I at a point P at a distance r from the current element is:
dB = (μ₀ / 4π) (Idl × r) / r³
To obtain the total field at P, we must integrate this vector expression over the entire length of the conductor.

The Biot-Savart law describes how current-carrying wires produce magnetic fields around them. It states that a tiny segment of wire (dl) carrying current contributes a small amount to the overall magnetic field (dB) felt at a point P in space, determined by the distance from the wire (r) and the direction of the current. To find the total magnetic field, we need to combine (integrate) the contributions from all segments of the wire.

Consider a group of people standing in a long line with strings, which represents the wire with current. Each person can pull on the string to create waves (like magnetic fields). To figure out how strong the wave is at a distance away, you take into account how much each person pulls and from how far they are, adding it all together to find the result.

The magnitude of the magnetic field due to a circular coil of radius R carrying a current I at an axial distance x from the centre is
B = (μ₀/2)(I/R²)
At the centre this reduces to
B = (μ₀I)/(2R).

This section discusses the magnetic field produced at the center of a circular coil when a current flows through it. The formula presented shows that the magnetic field depends on the current (I) and the radius of the coil (R). When measuring at the very center, the equation simplifies, indicating that the magnetic field strength increases with more current and decreases with larger coil radii.

Think about how the density of a crowd can create a stronger influence when standing together in a smaller area—like people tightly packed in a small space creating more noise. In this case, the 'noise' or magnetic field becomes stronger when the volume of current (like people) flowing through the coil is greater or the coil’s size is smaller.

Ampere’s Circuital Law: Let an open surface S be bounded by a loop C. Then the Ampere’s law states that ∫C B·dl = μ₀ I where I refers to the current passing through S.

Ampere's Circuital Law relates magnetic fields to the currents that produce them. Using this law, one can calculate the magnetic field around a closed loop (C) based on the total current (I) passing through that loop's area (S). It shows how magnetic fields form closed loops around current-carrying conductors, emphasizing the relationship between current and magnetic fields.

It’s like tracing the path of a river around a piece of land. The current flowing through the river affects how it flows and shapes the land around it, just as the electric current shapes the magnetic fields around wires.

Parallel currents attract and anti-parallel currents repel.

This principle observes the interaction between two parallel conductors carrying electric currents. When both currents flow in the same direction, they attract each other. When they flow in opposite directions, they repel each other. This behavior demonstrates how currents influence magnetic fields and subsequently interact with one another.

Picture two friends pulling on a tether rope while facing each other. If they both pull (like parallel currents), they come closer together. If one lets go while the other pulls (like anti-parallel currents), they push away.

A planar loop carrying a current I, having N closely wound turns, and an area A possesses a magnetic moment m where,
m = NIA.

The magnetic moment (m) of a loop is a measure of its ability to produce a magnetic field. It is influenced by the current (I) flowing through the loop, the number of turns (N), and the area (A) of the loop. This relationship highlights how loops with different configurations or currents can create varying strengths of magnets.

Envision a spinning playground merry-go-round. The faster it spins (like more current), the stronger the feeling when you stand near it (its magnetic moment). A loop with more layers (more turns) will create an even greater effect.

When this loop is placed in a uniform magnetic field B, the force F on it is: F = 0
And the torque on it is,
τ = m × B.

When the loop enters a uniform magnetic field, it experiences a torque (τ) that tends to rotate it. The torque's magnitude is based on the product of the magnetic moment (m) and the magnetic field (B). This interaction demonstrates how magnetic fields can affect physical objects that carry electrical currents.

Imagine a wind turbine catching the wind. The wind pushes the blades (akin to the magnetic force), causing the turbines to spin or rotate (similar to the torque on the magnetic loop).

In a moving coil galvanometer, this torque is balanced by a counter-torque due to a spring, yielding
kφ = NIAB
where φ is the equilibrium deflection and k the torsion constant of the spring.

The moving coil galvanometer measures current by using a coil that spins in a magnetic field. The torque from the magnetic field is countered by the tension in a spring, reaching an equilibrium at a certain angle (φ). This relationship allows for accurate measurements of electrical current through observable deflections.

Think of a balance beam scale, where weights on one side counterbalance the other side. The current's effect on the coil is like adding weight on one side, while the spring tension acts as the other side's weight, keeping it in check.

A moving coil galvanometer can be converted into an ammeter by introducing a shunt resistance r of small value in parallel. It can be converted into a voltmeter by introducing a resistance of large value in series.

To use a galvanometer as an ammeter, which measures current, a small shunt resistor is added in parallel with it, allowing most current to bypass the sensitive galvanometer. To measure voltage instead, a high resistance is connected in series to ensure the galvanometer does not draw too much current.

This is like using a big bucket (ammeter) to catch most of the rainwater while a small cup (galvanometer) catches just a bit of it for a more sensitive readout. When measuring voltage (voltmeter), the big bucket goes on the ground, not collecting rain as much.

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