IBDP Physics | D. Fields Concepts Summary

⚡️ Gravitational Fields - Quick Revsion

1. Kepler’s Three Laws of Orbital Motion

These laws describe the motion of planets around the Sun:

• First Law (Law of Ellipses)
  Each planet moves in an elliptical orbit with the Sun at one focus.

• Second Law (Law of Equal Areas)
  A line joining a planet and the Sun sweeps out equal areas in equal intervals of time.
  → This means that a planet moves faster when it is closer to the Sun and slower when it is farther away.

• Third Law (Law of Periods)
  The square of the orbital period $T$ of a planet is proportional to the cube of the semi-major axis $r$ of its orbit:

  $$T^2 \propto r^3$$

2. Newton’s Universal Law of Gravitation

This law quantifies the force of attraction between two point masses:

Where:

• $F$ = gravitational force
• $G$ = universal gravitational constant ($6.674 \times 10^{-11}\, \text{Nm}^2/\text{kg}^2$)
• $m_1, m_2$ = masses of the two bodies
• $r$ = distance between the centers of the masses

This law assumes that bodies are treated as point masses — that is, all their mass is concentrated at a single point.

3. Conditions for Treating Extended Bodies as Point Masses

An extended body (one with size and shape) can be approximated as a point mass when:

• Its size is much smaller compared to the distance between interacting bodies.
• Its mass is symmetrically distributed (especially for spherical bodies).
• We are interested only in motion or interaction far from the body itself (e.g., Earth–Sun system).

4. Gravitational Field Strength ($g$)

Gravitational field strength at a point is defined as the force per unit mass experienced by a small test mass:

Where:

• $g$ = gravitational field strength (in N/kg)
• $M$ = mass of the object creating the field (e.g., Earth)
• $r$ = distance from the center of the mass

$g$ decreases with distance from the mass and is directed toward the mass.

5. Gravitational Field Lines

• Gravitational field lines show the direction and strength of the field around a mass.
• They point towards the center of mass because gravity is always attractive.
• The closer the lines, the stronger the field.
• For a spherical mass (like a planet), field lines are radial and symmetric.

⚡️ Electric & Magnetic Fields – Quick Revision

🧲 Electric Forces & Charges

• Types of Charges

  • Positive $(+)$ and Negative $(-)$
  • Like charges repel, unlike charges attract

• Coulomb’s Law

  $$F = k \frac{q_1 q_2}{r^2} \quad \text{where} \quad k = \frac{1}{4\pi \varepsilon_0}$$

  Explains force between point charges.

🧮 Charge Properties

• Conservation of Charge
  Electric charge is neither created nor destroyed.

• Quantization of Charge
  Charge comes in discrete units, multiples of

  $$e = 1.6 \times 10^{-19} \, \text{C}$$

• Millikan’s Oil Drop Experiment
  Provided experimental evidence for charge quantization.

🔁 Charge Transfer Methods

• Friction
  Electrons are transferred between objects when rubbed.

• Electrostatic Induction
  Rearrangement of charges due to nearby charged objects.

• Contact & Grounding
  Direct transfer or neutralization of charge via contact with another object or the Earth.

🌐 Electric Field Concepts

• Electric Field Strength

  $$E = \frac{F}{q}$$

• Uniform Electric Field

  $$E = \frac{V}{d}$$

🧭 Field Lines

• Electric Field Lines
  Show direction of field: point away from positive and toward negative.

• Field Line Density
  More lines per unit area = stronger field.

• Magnetic Field Lines
  Point from North to South outside the magnet.

Motion of charge in fields - Quick Revsion

Motion of a Charged Particle in a Uniform Electric Field

When a charged particle ($q$) enters a uniform electric field ($\vec{E}$), it experiences a constant force $\vec{F}_E = q\vec{E}$. This leads to:

• Constant Acceleration: $\vec{a} = \frac{q\vec{E}}{m}$
• Trajectory: Straight line (if initial velocity is parallel to $\vec{E}$) or parabola (if initial velocity has a perpendicular component).

Motion of a Charged Particle in a Uniform Magnetic Field

A charged particle ($q$) moving with velocity $\vec{v}$ in a uniform magnetic field ($\vec{B}$) experiences a magnetic force $\vec{F}_B = q(\vec{v} \times \vec{B})$. Key aspects:

• Force Direction: Perpendicular to both $\vec{v}$ and $\vec{B}$ (right-hand rule for positive charges).
• Constant Speed: Magnetic force does no work.
• Trajectory:
  • Straight line if $\vec{v} \parallel \vec{B}$ or $\vec{v} \parallel -\vec{B}$.
  • Circular path with radius $r = \frac{mv}{qB}$ if $\vec{v} \perp \vec{B}$. Angular velocity $\omega = \frac{qB}{m}$.
  • Helix if $\vec{v}$ has components parallel and perpendicular to $\vec{B}$.

Motion in Perpendicular Electric and Magnetic Fields

The Lorentz force $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ governs the motion.

• Velocity Selector: Undeflected motion when $v = \frac{E}{B}$ (electric and magnetic forces balance).
• Mass Spectrometer: Utilizes velocity selection and circular motion to determine the charge-to-mass ratio.

Force on a Moving Charge in a Magnetic Field

Magnitude: $F = qvB\sin\theta$ Direction: Right-hand rule, perpendicular to $\vec{v}$ and $\vec{B}$.

Force on a Current-Carrying Conductor in a Magnetic Field

Magnitude: $F = BIL\sin\theta$ where $\theta$ is the angle between current direction and $\vec{B}$. Direction: Right-hand rule, perpendicular to current and $\vec{B}$.

Force per Unit Length Between Parallel Wires

$\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}$

• Attractive: Currents in the same direction.
• Repulsive: Currents in opposite directions.

Fields Revision Questions

Motion of a Charged Particle in Fields

Question 1: A proton enters a uniform electric field directed vertically upwards. Describe the subsequent motion of the proton.

Answer: The proton, being positively charged, will experience a constant upward force ($\vec{F}_E = q\vec{E}$). This will cause the proton to accelerate upwards in the direction of the electric field. If it had an initial horizontal velocity, its trajectory would be a parabola curving upwards, similar to projectile motion under gravity.

Question 2: An electron moves horizontally into a uniform magnetic field directed into the page. Sketch the path of the electron.

Answer: The electron, being negatively charged, will experience a magnetic force given by $\vec{F}_B = q(\vec{v} \times \vec{B})$. Using the left-hand rule (or the right-hand rule with the opposite direction for the force), the force will be directed downwards. This force is always perpendicular to the velocity, causing the electron to move in a circular path in the clockwise direction.

Question 3: A charged particle moves undeflected through a region containing both a uniform electric field and a uniform magnetic field, which are perpendicular to each other and also perpendicular to the velocity of the particle. What can you say about the magnitudes of the electric and magnetic forces on the particle?

Answer: For the particle to move undeflected, the net force on it must be zero. This means the electric force ($\vec{F}_E = q\vec{E}$) and the magnetic force ($\vec{F}_B = q(\vec{v} \times \vec{B})$) must be equal in magnitude and opposite in direction. Therefore, $qE = qvB$, which simplifies to $E = vB$.

Forces in Magnetic Fields

Question 4: A positive charge moves at an angle of 30° to a uniform magnetic field. How would the magnitude of the magnetic force on the charge change if the angle were increased to 90°?

Answer: The magnitude of the magnetic force is given by $F = qvB\sin\theta$. When $\theta = 30^\circ$, $F_1 = qvB\sin(30^\circ) = 0.5qvB$. When $\theta = 90^\circ$, $F_2 = qvB\sin(90^\circ) = qvB$. Therefore, the magnitude of the magnetic force would double.

Question 5: A straight wire carrying a current is placed in a uniform magnetic field. Under what condition will the magnetic force on the wire be maximum? Under what condition will it be zero?

Answer: The magnitude of the magnetic force on a current-carrying wire is $F = BIL\sin\theta$, where $\theta$ is the angle between the current direction and the magnetic field.

• The force will be maximum when $\sin\theta$ is maximum, which occurs when $\theta = 90^\circ$. This means the wire is perpendicular to the magnetic field.
• The force will be zero when $\sin\theta$ is zero, which occurs when $\theta = 0^\circ$ or $\theta = 180^\circ$. This means the wire is parallel or anti-parallel to the magnetic field.

Force Between Parallel Wires

Question 6: Two long, straight parallel wires carry currents in the same direction. Describe the nature of the force between the wires and explain why this force arises.

Answer: The force between the wires is attractive. This arises because the current in the first wire produces a magnetic field around it. According to the right-hand rule, this magnetic field will exert a force on the current-carrying second wire. Since the currents are in the same direction, applying the right-hand rule again shows that this force on the second wire is directed towards the first wire. By Newton’s third law, the first wire experiences an equal and opposite (attractive) force from the magnetic field produced by the second wire.

Question 7: Two long, straight parallel wires are separated by a distance of 5 cm and carry currents of 2 A and 3 A in opposite directions. Calculate the force per unit length between the wires.

Answer: The force per unit length between the parallel wires is given by $\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}$.

Given: $I_1 = 2 \, \text{A}$, $I_2 = 3 \, \text{A}$, $r = 0.05 \, \text{m}$, and $\mu_0 = 4\pi \times 10^{-7} \, \text{T m A}^{-1}$.

$\frac{F}{L} = \frac{(4\pi \times 10^{-7} \, \text{T m A}^{-1}) \times (2 \, \text{A}) \times (3 \, \text{A})}{2\pi \times (0.05 \, \text{m})}$ $\frac{F}{L} = \frac{12\pi \times 10^{-7}}{0.1\pi} \, \text{N m}^{-1}$ $\frac{F}{L} = 120 \times 10^{-7} \, \text{N m}^{-1} = 1.2 \times 10^{-5} \, \text{N m}^{-1}$ Since the currents are in opposite directions, the force is repulsive.