Electromagnetism
Module 6: Particles and medical physics6.3 Electromagnetism |
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| 6.3.1 |
Magnetic fields
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| 6.3.2 |
Motion of charged particles
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| 6.3.3 |
Electromagnetism
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1. Magnetic fields:
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a) Introduction to Magnetic Fields
- Magnetic fields are invisible regions around a magnetic material or a moving electric charge where magnetic forces can be experienced.
- ⇒ Sources of Magnetic Fields:
- Moving Charges: When charges move, they create magnetic fields. For example, current in a wire generates a magnetic field around it.
- Permanent Magnets: These have atomic currents due to electron motion and spin, which create a persistent magnetic field.
- Figure 1 Sources of magnetic fields
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b) Magnetic Field Lines
- Magnetic field lines are visual tools used to represent the strength and direction of magnetic fields.
- Figure 2 Magnetic field lines north to south pole
- ⇒ Properties of Magnetic Field Lines:
- They originate from the north pole of a magnet and end at the south pole (outside the magnet).
- Inside the magnet, they complete the loop, running from the south pole to the north pole.
- The closer the field lines, the stronger the magnetic field.
- They never intersect each other.
- Field lines can be mapped using iron filings or a small compass placed in the magnetic field.
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c) Magnetic Field Patterns
- Magnetic fields have distinct patterns depending on the source:
- Figure 3 Magnetic field lines patterns
- ⇒ Long Straight Current-Carrying Conductor:
- The magnetic field forms concentric circles around the conductor.
- The direction of the field can be determined using the right-hand rule: Thumb in the direction of the current, and fingers curl in the direction of the field.
- ⇒ Flat Coil (Circular Loop):
- The magnetic field is similar to that of a small bar magnet.
- The field lines converge at the center, making the field strong and uniform at the center.
- ⇒ Long Solenoid:
- A solenoid is a long coil of wire with multiple loops.
- The field inside the solenoid is strong and nearly uniform, resembling the field of a bar magnet.
- Outside the solenoid, the field is weaker and spreads out.
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d) Fleming’s Left-Hand Rule:
- Fleming’s Left-Hand Rule is used to determine the direction of the force on a current-carrying conductor in a magnetic field.
- Figure 4 Fleming’s Left-Hand Rule
- – Rule: Stretch your thumb, forefinger, and middle finger mutually perpendicular to each other.
- – Thumb: Direction of force (motion)
- – Forefinger: Direction of magnetic field
- – Middle finger: Direction of current
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e) I) Force on a Current-Carrying Conductor:
- When a conductor carrying current is placed in a magnetic field, it experiences a force due to the interaction between the magnetic field and moving charges in the conductor.
- ⇒ Formula:
- [math]F = BIL \sin \theta[/math]
- Where:
- – F = Force (Newtons)
- – B = Magnetic flux density (Tesla)
- – I = Current (Amperes)
- – L = Length of the conductor in the field (meters)
- – [math]\theta[/math]= Angle between the magnetic field and current
- Maximum Force: When [math]\theta = 90^\circ[/math] (current is perpendicular to the magnetic field).
- No Force: When [math]\theta = 0^\circ[/math] (current is parallel to the magnetic field).
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II) Techniques to Determine Magnetic Flux Density:
- Magnetic flux density (B) is measured in the region between the poles of a magnet using the following method:
- ⇒ Setup:
- – Place a straight current-carrying wire in a uniform magnetic field between two magnetic poles.
- – Connect the wire to a circuit with a variable power supply.
- – Use a digital balance to measure the force on the wire.
- ⇒ Procedure:
- – Adjust the current (I) through the wire.
- – Measure the force (F) using the digital balance.
- – Use the length (L) of the wire in the magnetic field and the measured force to calculate B:
- [math]B = \frac{F}{IL}[/math]
- ⇒ Outcome:
- This method gives the uniform magnetic flux density between the poles of the magnet.
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f) Magnetic Flux Density:
- Definition: Magnetic flux density (B) represents the amount of magnetic flux passing through a unit area perpendicular to the field.
- Figure 5 Magnetic flux density
- Unit: The SI unit is Tesla (T).
- [math]1 \, \text{Tesla (T)} = 1 \, \text{Newton per Ampere-meter (N/A} \cdot \text{m)}[/math]
- ⇒ Applications and Implications
- Practical Uses:
- – Electromagnetic devices (motors, generators, transformers) rely on magnetic fields
- – Medical imaging techniques like MRI use strong magnetic fields.
- Importance in Physics:
- – Understanding magnetic flux density and forces is essential in designing efficient electrical machines and understanding natural phenomena such as Earth’s magnetic field.
- By understanding these principles, we can map and manipulate magnetic fields for numerous practical applications in technology and science.
2. Motion of Charged Particles in Magnetic and Electric Fields
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a) Force on a Charged Particle in a Magnetic Field:
- When a charged particle moves through a magnetic field, it experiences a force. If the motion is at right angles to the magnetic field ([math]θ = 90^0[/math]), the force is given by:
- [math]F = BQv[/math]
- Where:
- – F = Force on the particle (Newtons)
- – B = Magnetic flux density (Tesla)
- – Q = Charge of the particle (Coulombs)
- – v = Velocity of the particle (m/s)
- ⇒ Direction of the Force:
- The force is always perpendicular to both the velocity of the particle and the magnetic field. The direction is determined by the Right-Hand Rule (for positive charges) or Left-Hand Rule (for negative charges).
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b) Charged Particles in a Uniform Magnetic Field
- When a charged particle enters a uniform magnetic field at right angles, it experiences a continuous perpendicular force, which causes it to move in a circular path.
- ⇒ Reason for Circular Motion:
- The magnetic force acts as the centripetal force:
- [math]F = \frac{mv^2}{r}[/math]
- Relationship Between Radius and Velocity:
- – Equating [math]F = BQv[/math] with the centripetal force:
- [math]r = \frac{mv}{BQ}[/math]
- Where:
- – r = Radius of the circular path (meters)
- – m = Mass of the particle (kg)
- – v = Velocity of the particle (m/s)
- – B = Magnetic flux density (Tesla)
- – Q = Charge of the particle (Coulombs)
- Circular Motion:
- – Higher velocity or mass leads to a larger radius.
- – Stronger magnetic field (B) or larger charge (Q) results in a smaller radius.
- c) Charged Particles in Combined Electric and Magnetic Fields
- When a charged particle moves through a region containing both electric (E) and magnetic (B) fields, it experiences forces from both fields:
- Electric Force ([math]F_E [/math]):
- [math]F_E = QE [/math]
- Magnetic Force ():
- [math]F_B = BQv [/math]
- The net force is the vector sum of these two forces, and the particle’s motion depends on the relative strengths and directions of E and B.
- ⇒ Velocity Selector:
- A velocity selector uses perpendicular electric and magnetic fields to allow only particles with a specific velocity to pass through.
- Principle of Operation:
- – For a charged particle moving through a velocity selector, the electric and magnetic forces act in opposite directions. For particles to pass through unaffected, the forces must cancel each other:
- [math]QE = BQv [/math]
- Simplifying:
- [math] v = \frac{E}{B}[/math]
- Where:
- – v = Velocity of the particle (m/s)
- – E = Electric field strength (V/m)
- – B = Magnetic flux density (Tesla)
- Applications:
- – Particles with [math] v = \frac{E}{B}[/math] pass through without deflection, while others are deflected and removed.
- – Used in mass spectrometers to separate ions by velocity.
- ⇒ Applications in Real Life
- Circular Motion of Charged Particles:
- – Found in devices like cyclotrons and synchrotrons, which accelerate particles for research in nuclear physics.
- – Helps analyze atomic and subatomic particles.
- Velocity Selector:
- – Used in mass spectrometry to analyze the composition of substances by separating ions based on mass-to-charge ratio.
- These principles underpin many scientific and technological advancements, including medical imaging, particle physics, and material science.
3. Electromagnetism
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a) Magnetic Flux (ϕ)
- Magnetic flux quantifies the total magnetic field passing through a given surface.
- ⇒ Formula for Magnetic Flux:
- [math]ϕ = BA cosθ[/math]
- Where:
- – ϕ = Magnetic flux (Webers, Wb)
- – B = Magnetic flux density (Tesla, T)
- – A = Area of the surface (m²)
- – θ = Angle between the magnetic field (B) and the normal to the surface
- ⇒ Unit of Magnetic Flux:
- The unit of magnetic flux is the weber (Wb).
- [math]1 \, weber = 1 \, Tesla meter^2[/math]
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b) Magnetic Flux Linkage
- Magnetic flux linkage refers to the total magnetic flux passing through all the turns of a coil.
- Formula:
- [math]Nϕ = NBA cosθ [/math]
- Where:
- – N = Number of turns in the coil
- – ϕ = Magnetic flux through a single loop
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c) Faraday’s Law of Electromagnetic Induction:
- Faraday’s Law states that a change in magnetic flux linkage induces an electromotive force (e.m.f.) in a circuit.
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I) Formula for Induced e.m.f.:
- [math] ε = -\frac{∆(Nϕ)}{Δt}[/math]
- Where
- – ε = Induced e.m.f. (Volts)
- – N = Magnetic flux linkage
- – Δt = Time interval for the change
- The negative sign in the formula comes from Lenz’s Law, which states that the direction of the induced e.m.f. opposes the change in flux causing it.
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II) Techniques to Investigate Magnetic Flux Using Search Coils
- Search Coil: A small coil of wire used to measure magnetic flux changes.
- Procedure:
- – Connect the search coil to a sensitive voltmeter or data logger.
- – Move the coil through a magnetic field or vary the field strength.
- – Observe the induced voltage to calculate changes in magnetic flux using Faraday’s Law.
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e) Simple a.c. Generator
- An a.c. generator converts mechanical energy into electrical energy by electromagnetic induction.
- ⇒ Components:
- A coil (rotor) rotates in a uniform magnetic field.
- Brushes and slip rings maintain electrical contact with the external circuit.
- ⇒ Working Principle:
- As the coil rotates, the magnetic flux linkage through it changes periodically, inducing an alternating e.m.f.
- [math] ε = -\frac{∆(Nϕ)}{Δt}[/math]
- – The output voltage varies sinusoidally with time.
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f) I) Transformers
- Transformers change the voltage level in an alternating current (a.c.) circuit by electromagnetic induction.
- ⇒ Ideal Transformer Formula
- For an ideal transformer:
- [math]\frac{N_s}{N_p} = \frac{V_s}{V_p} = \frac{I_p}{I_s} [/math]
- Where:
- – [math]N_s , N_p [/math]: Number of turns in secondary and primary coils
- - [math]V_s , V_p [/math]: Secondary and primary voltages (Volts)
- - [math]I_s , I_p [/math]: Secondary and primary currents (Amperes)
- Step-up transformer: Increases voltage ([math]N_s > N_p [/math])
- Step-down transformer: Decreases voltage ([math]N_s < N_p [/math])
- ⇒ Structure of a Transformer
- Core: Laminated iron core reduces energy loss due to eddy currents.
- Windings: Primary and secondary windings insulated to minimize leakage currents.
- ⇒ Efficiency of a Transformer
- For an ideal transformer:
- [math] \text{Power Input} = \text{Power Output} → V_p I_p = V_s I_s [/math]
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II) Techniques to Investigate Transformers
- Procedure:
- – Connect the primary coil to an a.c. power supply.
- – Attach a voltmeter across the primary and secondary coils to measure input and output voltages.
- – Adjust the number of turns in the secondary coil and observe the voltage changes.
- – Use an ammeter to measure current and confirm [math]V_p I_p = V_s I_s [/math] for efficiency.
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g) Applications of Electromagnetism
- Electric Generators: Production of electricity in power plants.
- Transformers: Used to step up/down voltage in power transmission.
- Magnetic Induction: Utilized in wireless charging systems and electric motors.
- Search Coils: Employed in detecting magnetic anomalies and conducting flux experiments.
- These principles are vital for modern electrical systems, power distribution, and electromotive devices