Physics | Electric and Magnetic Fields

Welcome to Electric and Magnetic Fields!

Hello future physicist! This chapter dives into one of the most fundamental and fascinating concepts in all of Physics: how electric charges and magnetic fields interact. Don't worry if the formulas look complicated at first—we will break down every concept into simple, manageable steps.

You've already met forces and mechanics. Now, we are learning about forces that act without touching, caused by invisible fields. Understanding these fields is essential because they govern everything from how your TV screen works to how particle accelerators guide beams of atoms! Let's get started.

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Section 1: Electric Fields Revisited – The Force on a Charge

1.1 Defining Electric Field Strength (\(E\))

An Electric Field is a region where a charged particle experiences a force. We define the strength of this field based on the force experienced by a single unit of positive charge.

Key Equation: Force on a Charge

The force (\(F\)) exerted on a charge (\(q\)) placed in an electric field of strength \(E\) is given by:

\(F = Eq\)

Where:

• \(F\) is the Force (N)
• \(q\) is the charge (C)
• \(E\) is the Electric Field Strength (N C\(\text{^{-1}}\) or V m\(\text{^{-1}}\))

Analogy: Think of the Earth's gravitational field. The field strength (\(g\)) causes a force (\(F\)) on a mass (\(m\)) such that \(F = mg\). Electric fields work the same way, just for charge instead of mass!

1.2 Uniform Electric Fields (Parallel Plates)

In many A-Level problems, we deal with fields between two parallel metal plates connected to a voltage supply. These fields are uniform, meaning the field strength \(E\) is constant everywhere between the plates (except near the edges).

If a potential difference (\(V\)) is applied across two plates separated by distance (\(d\)):

\(E = \frac{V}{d}\)

Motion of Charges in a Uniform E-Field

When a charged particle (like an electron) enters the field, it experiences a constant force.

• If the particle moves parallel to the field lines, it simply speeds up or slows down (linear acceleration).
• If the particle moves perpendicular to the field lines (like entering from the side), the constant electric force acts like gravity in projectile motion. The path curves into a parabola.

Quick Review: The motion is analyzed by treating the forces separately: uniform motion (constant velocity) perpendicular to \(F\), and accelerated motion parallel to \(F\).

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Section 2: Magnetic Fields – The Mysterious Force

2.1 Introducing Magnetic Flux Density (\(B\))

Just as \(E\) describes the strength of an electric field, we use Magnetic Flux Density (\(B\)) to describe the strength of a magnetic field.

• The unit for \(B\) is the Tesla (T).

2.2 Force on a Charged Particle in a Magnetic Field

A key difference from electric fields: a magnetic field only exerts a force on a charged particle if the particle is moving, and only if its velocity vector has a component perpendicular to the field lines.

When a charge \(q\) moves at velocity \(v\) perpendicular to a magnetic field \(B\), the force \(F\) is calculated using:

\(F = Bqv\)

Where:

• \(F\) is the Force (N)
• \(B\) is the Magnetic Flux Density (T)
• \(q\) is the charge (C)
• \(v\) is the velocity (m s\(\text{^{-1}}\))

Important Points to Remember:

1. If the particle is stationary (\(v=0\)), \(F=0\).
2. If the particle moves parallel to the field lines, \(F=0\).
3. The force \(F\) is always perpendicular to both the velocity (\(v\)) and the field (\(B\)).

2.3 Determining Direction: Fleming's Left-Hand Rule (FLHR)

Since magnetic force is a vector (it has direction), we must use a rule to figure out where the force points. This rule is crucial for magnetic field problems!

Use your left hand and spread your thumb, forefinger, and middle finger so they are all mutually perpendicular (at 90° to each other):

Forefinger: Field (Direction of B, North to South)
Middle finger: Intensity (Current, or direction of positive charge movement)
Thumb: Thrust (Force or Motion)

Memory Aid: F-B-I (Force, Field, Current/Velocity).

A Note on Electrons (Negative Charges):

The direction of current (\(I\)) is defined as the flow of positive charge. If you are dealing with an electron moving left, the conventional current direction (\(I\)) is moving right. When using FLHR for electrons, point your middle finger in the direction opposite to the electron's actual motion!

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Section 3: The Motor Effect – Force on a Conductor

3.1 Why Current-Carrying Wires Feel a Force

A current (\(I\)) in a wire is simply a massive flow of charged particles. If a single charge feels a force (\(F=Bqv\)), then a wire carrying current must also feel a cumulative force. This is the principle behind the Motor Effect.

For a straight wire of length \(L\) carrying a current \(I\) placed perpendicular to a magnetic field \(B\), the total force \(F\) is:

\(F = BIL\)

Where:

• \(F\) is the Force (N)
• \(B\) is the Magnetic Flux Density (T)
• \(I\) is the Current (A)
• \(L\) is the length of the conductor inside the field (m)

Did you know? This equation defines the unit of the Tesla. One Tesla is the magnetic flux density that produces a force of 1 N on a 1 m wire carrying 1 A of current perpendicular to the field.

3.2 Real World Application: Motors

This force \(F = BIL\) is what makes electric motors spin! A coil of wire carrying a current is placed in a magnetic field. The force pushes one side of the coil up and the other side down, creating a turning effect (torque).

Common Mistake to Avoid:

If the wire is not perpendicular to the field, you must use the component of the current that is perpendicular to the field lines. If the angle between \(I\) and \(B\) is \(\theta\), the full formula is \(F = BIL \sin \theta\). If they are parallel (\(\theta=0\)), \(\sin \theta=0\) and \(F=0\).

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Section 4: Advanced Motion – Circular and Combined Fields

4.1 Circular Motion in a Magnetic Field

We established that the magnetic force \(F\) on a moving charge is always perpendicular to its velocity \(v\).

What happens when a force is always perpendicular to motion? It doesn't change the speed of the object; it only changes the direction. This results in uniform circular motion.

When a charged particle enters a uniform magnetic field perpendicularly, it follows a circular path. To analyze this, we equate the magnetic force to the centripetal force required for circular motion:

Step-by-Step Derivation of Radius \(r\)

1. Magnetic Force: The force causing the turn is the magnetic force: $$F_{B} = Bqv$$
2. Centripetal Force: For circular motion, a centripetal force is required: $$F_{c} = \frac{mv^2}{r}$$
3. Equate Forces: Since \(F_{B} = F_{c}\): $$Bqv = \frac{mv^2}{r}$$
4. Solve for Radius \(r\): Rearrange the equation to find the radius of the circle: $$r = \frac{mv}{Bq}$$

This equation (\(r = \frac{mv}{Bq}\)) is fundamental. It shows that:

• Faster particles (\(v\)) have larger radii (\(r\)).
• Heavier particles (\(m\)) have larger radii (\(r\)).
• Stronger fields (\(B\)) or larger charges (\(q\)) result in smaller radii (\(r\)).

This principle is used in instruments like mass spectrometers to separate ions based on their mass-to-charge ratio (\(m/q\)).

4.2 Velocity Selector (Combined E and B Fields)

Imagine an apparatus where a uniform electric field (\(E\)) and a uniform magnetic field (\(B\)) are set up perpendicular to each other, and a beam of charged particles is fired through them. This arrangement is called a velocity selector.

By carefully setting the directions, we can make the forces cancel out:

• The Electric Force (\(F_E = Eq\)) pushes the particle one way.
• The Magnetic Force (\(F_B = Bqv\)) pushes the particle the opposite way.

If the forces are equal and opposite, the particle passes straight through undeflected.

Condition for Zero Deflection:

$$\text{Force Electric} = \text{Force Magnetic}$$ $$F_E = F_B$$ $$Eq = Bqv$$

We can cancel \(q\) (charge) from both sides and solve for the velocity (\(v\)) of the undeflected particles:

$$v = \frac{E}{B}$$

The Power of the Velocity Selector: Only particles with this specific velocity \(v\) will pass straight through the selector. All other particles (faster or slower) will be deflected by the stronger force acting on them. This allows scientists to select particles moving at a precise speed.