DP IB Physics: SL
D. Fields
D.3 Motion in electromagnetic fields
DP IB Physics: SLD. FieldsD.3 Motion in electromagnetic fieldsUnderstandings Standard level and higher level: 8 hours |
|
|---|---|
| a) | The motion of a charged particle in a uniform electric field |
| b) | The motion of a charged particle in a uniform magnetic field |
| c) | The motion of a charged particle in perpendicularly orientated uniform electric and magnetic fields |
| d) | The magnitude and direction of the force on a charge moving in a magnetic field as given by [math]F = qvB \sin\theta[/math] |
| e) | The magnitude and direction of the force on a current-carrying conductor in a magnetic field as given by [math]F = B I L \sin\theta[/math] |
| f) |
The force per unit length between parallel wires as given by [math]\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}[/math] Where r is the separation between the two wires. |
-
a) Motion of a Charged Particle in a Uniform Electric Field
- A uniform electric field exerts a constant force on a charged particle, causing it to accelerate in a straight line.
- ⇒ Force on the Particle:
- The electric force on a charged particle is given by:
- [math]F = qE[/math]
- Where:
- – F = electric force (in newtons, N)
- – q = charge of the particle (in coulombs, C)
- – E = electric field strength (in N/C or V/m)
- Figure 1 Force on the charge particle
- ⇒ Nature of Motion:
- The acceleration is constant, because the force is constant (from Newton’s 2nd law: F = ma ).
- The particle experiences linear uniformly accelerated motion.
- [math]a = \frac{F}{m} \\
a = \frac{qE}{m}[/math] - Where:
- – m = mass of the particle
- ⇒ Direction of Motion:
- Positive charge: accelerates in the direction of the electric field.
- Negative charge: accelerates opposite to the field
- Example – Between Parallel Plates:
- A charged particle enters the region between two parallel plates with a horizontal velocity.
- – Horizontally: continues at constant speed (no horizontal force).
- – Vertically: undergoes constant acceleration due to electric field.
- Figure 2 Motion of charge particle
- Result: Parabolic trajectory — similar to a projectile under gravity.
- Figure 3 Motion of charge particle between parallel plates
-
b) Motion of a Charged Particle in a Uniform Magnetic Field
- ⇒ Magnetic Force:
- The magnetic force on a moving charge is given by:
- [math]F = qvB \sin\theta
[/math] - Where:
- – F = magnetic force (N)
- – q = charge (C)
- – v = velocity of the particle (m/s)
- – B = magnetic field strength (T)
- – θ = angle between velocity and magnetic field
- Figure 4 Motion of charge particle in a uniform magnetic field
- ⇒ Velocity ⟂ Field (90°)
- If the particle moves perpendicular to the magnetic field:
- – The magnetic force acts perpendicular to the velocity at all times.
- – The particle moves in a circular path.
- – The force acts as a centripetal force, keeping the particle in circular motion.
- [math]qvB = \frac{mv^2}{r} \\
r = \frac{mv}{qB}[/math] - Where:
- – r = radius of circular path
- ⇒ Nature of Motion:
- The particle moves in a circle (if v ⟂ B)
- If the velocity has a component along the field, the particle moves in a helix or spiral
- ⇒ Direction of Force:
- Use the Right-Hand Rule:
- Point your fingers in the direction of v,
- Turn your palm toward B,
- Your thumb points in the direction of force on a positive
- For negative charges (like electrons), the force is opposite to your thumb.
| Scenario | Path | Force Direction | Motion Type |
|---|---|---|---|
| Electric Field (uniform) | Straight Line | Constant | Linear acceleration |
| Magnetic Field (uniform, ⟂ velocity) | Circular | Perpendicular to motion | Uniform circular motion |
| Magnetic Field (at an angle) | Helical | Changes direction | Helical path |
- ⇒ Differences
| Property | Electric Field | Magnetic Field |
|---|---|---|
| Force depends on | Charge q and Field E | Charge q, speed v, and field B |
| Direction of force | Along field (or opposite) | Perpendicular to velocity |
| Work done | Yes (changes speed) | No (changes direction only) |
| Motion | Accelerated (parabolic/linear) | Circular or helical |
-
c) Motion of a Charged Particle in Perpendicularly Oriented Uniform Electric and Magnetic Fields
- This is a crossed-field setup, where:
- A uniform electric field ([math]\vec{E}[/math]) and a uniform magnetic field ( [math]\vec{B}[/math]) are perpendicular to each other.
- A charged particle moves through both fields.
- This setup is called a velocity selector when the fields are tuned such that only particles with a certain velocity pass through undeflected.
- ⇒ Forces Acting on the Particle:
- The particle experiences two forces:
- Electric Force:
- [math]\vec{F}_E = q \vec{E} [/math]
- Direction: along the electric field for a positive charge, opposite for a negative charge.
- Magnetic Force:
- [math]\vec{F}_B = q \vec{v} \times \vec{B}[/math]
- Direction: perpendicular to both velocity and magnetic field.
- Use the Right-Hand Rule to find direction (for positive charges).
- If Both Forces Are Equal and Opposite:
- [math]qE = qvB \\ v = \frac{E}{B}[/math]
- This is the velocity selector condition: only particles with velocity [math]v = \frac{E}{B}[/math] experience no net force and move in a straight line.
- All other particles are deflected.
- ⇒ Trajectory of the Particle:
- If [math]v \neq \frac{E}{B}[/math], then:
- – The net force will cause curved motion.
- – The resulting path can be helical, circular, or parabolic, depending on initial direction and speed.
- ⇒ Application:
- Mass spectrometers
- Cathode ray tubes
- Charged particle filters
-
d) Force on a Charge Moving in a Magnetic Field:[math]F = q v B \sin \theta[/math]
- ⇒ Magnetic Force Formula:
- [math]F = q v B \sin \theta[/math]
- Where:
- – F = magnetic force (N)
- – q = charge (C)
- – v = speed of the particle (m/s)
- – B = magnetic field strength (T)
- – θ = angle between [math]\vec{v} \text{ and } \vec{B}[/math]
| Condition | Force |
|---|---|
| [math]\theta = 0^\circ \text{ or } 180^\circ[/math] | F=0 (motion parallel or anti-parallel to field; no deflection) |
| [math]\theta = 90^\circ [/math] | [math]F = q v B[/math](maximum force; circular motion) |
| θ= any other angle | [math]F = q v B \sin \theta[/math](particle follows a helical path) |
- ⇒ Direction of Force:
- Given by the Right-Hand Rule:
- – Point fingers in the direction of [math]\vec{v} \text{ (velocity)}[/math],
- – Curl toward [math]\vec{B}[/math],
- – Thumb points in the direction of the force for a positive
- – For negative charges (like electrons), the force is in the opposite direction to your thumb.
- Figure 5 Direction of motion by right hand rule
- ⇒ Example:
- If:
- – [math]q = 1.6 \times 10^{-19} \text{ C} \\
– v = 2.0 \times 10^{6} \frac{\text{m}}{\text{s}} \\
– B = 0.50 \text{ T} \\
– \theta = 90^\circ[/math] - Then:
- [math]F = q v B \\
F = (1.6 \times 10^{-19})(2.0 \times 10^{6})(0.50) \\
F = 1.6 \times 10^{-13} \text{ N}[/math] -
e) Force on a Current-Carrying Conductor in a Magnetic Field
- [math]F = BIL \sin \theta[/math]
- Where:
- – F = magnetic force (N)
- – B = magnetic field strength (tesla, T)
- – I = current (amperes, A)
- – L = length of the conductor inside the field (meters, m)
- – θ = angle between the conductor (current direction) and the magnetic field
- Figure 6 Force on a current-carrying conductor in a magnetic field
- ⇒ Explanation:
- A current in a conductor consists of moving charges (electrons).
- These moving charges experience a magnetic force when placed in a magnetic field.
- As a result, the whole wire experiences a force.
- ⇒ Special Cases:
| Case | Explanation | Force |
|---|---|---|
| [math]\theta = 90^\circ[/math] | Wire is perpendicular to the magnetic field | F=BIL (maximum) |
| [math]\theta = 0^\circ[/math] | Wire is parallel to the magnetic field | F=0 (no force) |
- Direction of Force – Right-Hand Rule:
- Point your fingers in the direction of current (I),
- Turn your palm toward the direction of the magnetic field (B),
- Your thumb points in the direction of the force on the wire.
- For positive current, use the thumb’s direction.
- Figure 7 Direction of force by right hand rule
- Practical Example:
- In electric motors, loops of wire inside a magnetic field experience a rotational force due to this principle.
- This force creates torque, which drives rotation.
-
f) Force Per Unit Length Between Two Parallel Current-Carrying Wires
- [math]\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2 \pi r}[/math]
- Where:
- - [math]\frac{F}{L}[/math]= force per unit length (N/m)
- – [math]\mu_0 = \text{Permeability of free space} = 4 \pi \times 10^{-7} \frac{\text{T} \cdot \text{m}}{\text{A}}[/math]
- – [math]I_1 , I_2[/math] = currents in the two wires (A)
- – r = separation between wires (m)
- ⇒ Explanation:
- A current-carrying wire produces a magnetic field around it.
- If another wire is placed nearby, it will experience a magnetic force due to the field of the first wire.
- This force depends on:
- – The magnitude of both currents,
- – The distance between the wires,
- – The direction of the currents.
- Figure 8 Magnetic force due to current-carrying wire
- ⇒ Attraction or Repulsion:
| Currents Direction | Force |
|---|---|
| Same direction | Attractive force |
| Opposite direction | Repulsive force |
- This is a direct result of the magnetic field lines created by the wires and how they interact.
- Direction of Force – Right-Hand Rule (for interaction):
- For wire 1: use the right-hand rule to find direction of magnetic field at wire 2.
- Use [math]F = I \cdot L \cdot B[/math] on wire 2 to find direction of force on it.
- Repeat for wire 2 acting on wire 1 — Newton’s Third Law applies: action = reaction.
- ⇒ Application: The Definition of the Ampere
- The ampere is defined based on this force:
- If two long, straight, parallel conductors placed 1 meter apart in vacuum each carry a current of 1 ampere, the force per meter between them is:
- [math]\frac{F}{L} = 2 \times 10^{-7} \text{ N/m}[/math]