9. Magnetic Fields A.md

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20.1 Concept of a Magnetic Field

Conceptual Understanding: What is a magnetic field, and what are the two primary sources that produce it? Answer: A magnetic field is a region where a magnetic force can be experienced. It is produced either by moving electric charges, such as in a current-carrying wire, or by permanent magnets.

Representation: Describe how magnetic field lines are used to represent a magnetic field and state what the properties of these lines represent. Answer: Magnetic field lines are used to visually represent magnetic fields. They show the direction of the magnetic field at different points in space. The direction of the field lines represents the direction a north pole would move if placed in the field. The density of the field lines (how close they are together) represents the strength of the magnetic field; closer lines indicate a stronger field.

20.2 Force on a Current-Carrying Conductor

Conceptual Understanding: When does a force act on a current-carrying conductor in a magnetic field? Answer: A force acts on a current-carrying conductor when it is in a magnetic field and the direction of the current is not parallel to the magnetic field lines. The force is strongest when the conductor is perpendicular to the magnetic field lines.

Formula Dictation and Application: State the formula that describes the force on a current-carrying conductor in a magnetic field, and calculate the force exerted on a 2-meter-long wire carrying a current of 3 A at a right angle to a magnetic field with a flux density of 0.5 T. Answer: The formula is . For a wire at a right angle () to the magnetic field, . Thus, .

Definition Dictation: Define magnetic flux density and explain its significance in relation to a current-carrying conductor in a magnetic field. Answer: Magnetic flux density is defined as the force per unit current per unit length exerted on a wire when it is placed at right angles to a magnetic field. It's a measure of the strength of the magnetic field, with the SI unit of tesla (T). It indicates how much force will be exerted on a wire of a given length carrying a certain current when placed in the field.

20.3 Force on a Moving Charge 6. Direction Determination: How can the direction of the force on a charge moving in a magnetic field be determined? Answer: The direction of the force on a charge moving in a magnetic field can be determined using Fleming's Left-Hand Rule. For a positive charge, point your thumb in the direction of the charge's velocity (v), your index finger in the direction of the magnetic field (B), and your middle finger will point in the direction of the force (F). For a negative charge, the direction of the force is opposite to that indicated by the middle finger.

Formula Dictation and Application: State the formula for the force on a moving charge in a magnetic field and calculate the force on a proton moving at \(2 \times 10^6\) m/s perpendicular to a magnetic field of strength \(1.5 \times 10^{-2}\) T. Answer: The formula is \(F = BQv \sin \theta\). For a proton moving perpendicular to the magnetic field, \(\sin \theta = 1\). Given \(Q = 1.6 \times 10^{-19}\) C (charge of a proton), \(B = 1.5 \times 10^{-2}\) T, and \(v = 2 \times 10^6\) m/s, the force is \(F = (1.5 \times 10^{-2} \text{ T}) \times (1.6 \times 10^{-19} \text{ C}) \times (2 \times 10^6 \text{ m/s}) \times 1 = 4.8 \times 10^{-15} \text{ N}\).

Hall Effect: Explain the origin of the Hall voltage and derive the expression for the Hall voltage \(VH = \frac{BI}{ntq}\), where \(t\) is the thickness.  Answer: The Hall voltage originates from the Hall effect, which occurs when a current-carrying conductor or semiconductor is placed in a magnetic field perpendicular to the current. This causes a buildup of charge on one side of the material, creating a voltage (the Hall voltage) across the material. The expression \(VH = \frac{BI}{ntq}\) is derived by considering the balance between the magnetic force and the electric force in the material, where \(B\) is the magnetic flux density, \(I\) is the current, \(n\) is the charge carrier density, \(t\) is the thickness of the conductor, and \(q\) is the charge of the carriers.

Hall Probe: Describe how a Hall probe can be used to measure magnetic flux density. Answer: A Hall probe measures magnetic flux density by exploiting the Hall effect. When a known current is passed through the probe placed in a magnetic field, a Hall voltage is produced across it, perpendicular to both the current and the magnetic field. The magnitude of this Hall voltage is directly proportional to the magnetic flux density. By measuring the Hall voltage and knowing the current, the thickness of the probe, and the charge carrier density, the magnetic flux density can be calculated.

Charged Particle Motion: Describe the motion of a charged particle moving in a uniform magnetic field when the particle's velocity is perpendicular to the field. Answer: When a charged particle moves in a uniform magnetic field with its velocity perpendicular to the field, it experiences a centripetal force that causes it to move in a circular path. The magnetic force provides the centripetal force necessary for circular motion. The radius of the circle depends on the velocity of the particle, its mass, the charge, and the strength of the magnetic field.

Velocity Selection: Explain how electric and magnetic fields can be used in velocity selection. Answer: Electric and magnetic fields can be used together in a velocity selector to select particles of a specific velocity. This is achieved by setting up perpendicular electric and magnetic fields such that the force due to the electric field on a charged particle is equal and opposite to the force due to the magnetic field. Only particles with a specific velocity, for which the electric force equals the magnetic force, can pass through the selector undeflected. The condition for this is \(v = \frac{E}{B}\), where \(v\) is the velocity, \(E\) is the electric field strength, and \(B\) is the magnetic field strength.

Section 20: Magnetic Fields

20.4 Magnetic Fields Due to Currents

Magnetic Field Patterns: Sketch the magnetic field patterns for the following current-carrying conductors: a long straight wire, a flat circular coil, and a long solenoid. Answer:

For a long straight wire, the magnetic field lines are concentric circles around the wire.

For a flat circular coil, the magnetic field lines resemble a pattern similar to that of a bar magnet with circular lines inside the coil and arcing lines outside the coil.

For a long solenoid, the field lines are parallel and close together inside the solenoid, indicating a uniform magnetic field, and loop around from one end to the other outside the solenoid.

Solenoid with a Ferrous Core: Explain how the magnetic field due to the current in a solenoid is affected by introducing a ferrous core. Answer: Introducing a ferrous core into a solenoid increases the magnetic field strength. This is because the ferrous material becomes magnetized and the magnetic domains within the core align with the field of the solenoid, effectively increasing the overall magnetic field through the addition of the core's magnetic field to that of the solenoid.

Forces Between Conductors: Explain the origin of the forces between current-carrying conductors and determine the direction of these forces. Answer: The forces between current-carrying conductors originate from the magnetic fields produced by the currents in each conductor. When two conductors carry currents in the same direction, they attract each other, and when they carry currents in opposite directions, they repel each other. The direction of the force can be determined using the right-hand grip rule: if you grip the conductor with your right hand with your thumb pointing in the direction of the current, your fingers curl in the direction of the magnetic field.

20.5 Electromagnetic Induction

Magnetic Flux: Define magnetic flux and state its units. Answer: Magnetic flux (\(\Phi\)) is defined as the product of the magnetic flux density (B) and the cross-sectional area (A) perpendicular to the direction of the magnetic flux density. It is measured in webers (Wb).

Magnetic Flux Calculation: Recall and use the equation \(\Phi = BA\) to calculate the magnetic flux through a surface area of 0.02 m² in a uniform magnetic field of 0.5 T. Answer: Given \(B = 0.5 \text{ T}\) and \(A = 0.02 \text{ m}^2\), the magnetic flux is \(\Phi = BA = (0.5 \text{ T})(0.02 \text{ m}^2) = 0.01 \text{ Wb}\).

Magnetic Flux Linkage: Understand and use the concept of magnetic flux linkage. Answer: Magnetic flux linkage refers to the total magnetic flux linking a coil and is given by the product of the magnetic flux through one loop of the coil and the total number of turns in the coil. It is a measure of how much magnetic field is 'enclosed' by the coil.

Electromagnetic Induction Experiments: Understand and explain experiments that demonstrate electromagnetic induction, including the factors affecting the magnitude of the induced e.m.f. Answer: Experiments that demonstrate electromagnetic induction typically involve moving a magnet into or out of a coil, changing the current in a nearby coil, or rotating a coil within a magnetic field. These changes in magnetic flux induce an e.m.f. in the coil. The magnitude of the induced e.m.f. depends on the rate of change of the magnetic flux, the number of turns in the coil, and the area of the coil.

Faraday’s and Lenz’s Laws: Recall and use Faraday’s and Lenz’s laws of electromagnetic induction. Answer: Faraday's law states that the induced e.m.f. in a circuit is directly proportional to the rate of change of magnetic flux linkage. Lenz's law states that the direction of the induced e.m.f. (and hence the induced current if the circuit is closed) is such that it opposes the change in magnetic flux that produced it. The equation for Faraday's law is \( \varepsilon = -N \frac{\Delta \Phi}{\Delta t} \), where \( \varepsilon \) is the induced e.m.f., \( N \) is the number of turns in the coil, and \( \frac{\Delta \Phi}{\Delta t} \) is the rate of change of magnetic flux.