AP Physics 2- 12.1 Magnetic Fields- Study Notes- New Syllabus
AP Physics 2- 12.1 Magnetic Fields- Study Notes
AP Physics 2- 12.1 Magnetic Fields- Study Notes – per latest Syllabus.
Key Concepts:
- Properties of a Magnetic Field
- Magnetic Behavior of Materials (Magnetic Dipoles)
- Magnetic Permeability of a Material
Properties of a Magnetic Field
A magnetic field is a region of space around a magnet, a current-carrying conductor, or a moving charged particle where magnetic forces can be observed. It is represented by the vector field \(\vec{B}\), also called the magnetic flux density.
Key Properties:
- Vector Quantity: A magnetic field has both magnitude and direction, represented by \(\vec{B}\).
- Direction: The direction of the magnetic field at a point is the direction in which the north pole of a test compass needle points.
Magnetic Field Lines:
- Magnetic field lines form closed continuous loops (from north pole to south pole outside a magnet, and from south to north inside).
- The density of field lines indicates the field’s strength: closer lines mean stronger magnetic field.
- Field lines never intersect.
No Monopoles: Magnetic poles always exist in pairs (north and south). Cutting a magnet produces two smaller magnets, each with both poles.
Source of Magnetic Field: Moving electric charges (currents) and changing electric fields produce magnetic fields (Ampère’s law and Maxwell’s equations).
Units: The SI unit of magnetic field \(\vec{B}\) is the Tesla (T). \( \mathrm{1 \, T = 1 \, N/(A \cdot m)} \).
Example :
A uniform magnetic field of strength \(\mathrm{0.5 \, T}\) is applied perpendicular to an area of \(\mathrm{0.02 \, m^2}\). Find the magnetic flux through the area.
▶️ Answer/Explanation
Step 1: Formula: \(\mathrm{\Phi_B = BA \cos \theta}\).
Step 2: Since the field is perpendicular, \(\theta = 0^\circ\), so \(\cos \theta = 1\).
Step 3: \(\mathrm{\Phi_B = (0.5)(0.02)(1) = 0.01 \, Wb}\).
Answer: Magnetic flux = \(\mathrm{0.01 \, Wb}\).
Magnetic Behavior of Materials (Magnetic Dipoles)
Magnetic behavior of a material is determined by the alignment of atomic magnetic dipoles inside the material. Each atom can be modeled as a small magnetic dipole due to the motion of electrons (orbital and spin). The collective arrangement of these dipoles dictates the material’s magnetic properties.
Key Concepts:
- Magnetic Dipoles: A magnetic dipole consists of a north and south pole separated by a small distance. At the atomic level, dipoles arise mainly from electron spin and orbital motion.
- Random Orientation: In the absence of an external magnetic field, dipoles may be randomly oriented, producing little or no net magnetization.
- Alignment in a Field: When an external magnetic field \(\vec{B}\) is applied, magnetic dipoles tend to align with the field, leading to magnetization.
Types of Magnetic Materials:
Diamagnetic Materials:
- Dipoles are not permanent; induced dipoles oppose the applied field.
- Very weak repulsion from the external magnetic field.
- Examples: Copper, Bismuth, Silver.
Paramagnetic Materials:
- Have permanent dipoles, but random orientation without a field.
- When a field is applied, dipoles partially align with the field.
- Weak attraction to magnetic fields.
- Examples: Aluminum, Platinum.
Ferromagnetic Materials:
- Strong permanent dipoles that can align spontaneously even without a field.
- Regions of aligned dipoles are called magnetic domains.
- External fields can make domains grow, producing strong magnetization.
- Can retain magnetization (hysteresis) after removing the external field.
- Examples: Iron, Nickel, Cobalt.
Antiferromagnetic and Ferrimagnetic Materials:
- Antiferromagnetic: Adjacent dipoles align in opposite directions, canceling each other’s effects (e.g., Manganese oxide).
- Ferrimagnetic: Adjacent dipoles align oppositely but unequally, leading to partial cancellation (e.g., Magnetite \(\mathrm{Fe_3O_4}\)).
Important Relation:
The magnetization \(\mathrm{\vec{M}}\) of a material is the net magnetic dipole moment per unit volume:
$ \mathrm{\vec{M} = \dfrac{\sum \vec{\mu}}{V}}$
where \(\mathrm{\vec{\mu}}\) is the magnetic dipole moment of each atom, and \(\mathrm{V}\) is the volume.
NOTE:
Earth’s magnetic field may be approximated as a magnetic dipole.
Example :
An iron nail is placed in a strong external magnetic field. Explain what happens to the magnetic dipoles inside the nail and describe the overall behavior of the nail.
▶️ Answer/Explanation
Step 1: Iron is ferromagnetic, meaning it has magnetic domains.
Step 2: In the absence of a field, the domains are randomly oriented, so the net magnetization is nearly zero.
Step 3: When placed in a strong external field, the domains align in the direction of the field, increasing in size and number.
Step 4: The iron nail becomes strongly magnetized and can act as a temporary magnet.
Answer: The nail’s magnetic dipoles align with the field, making the nail strongly magnetic.
Magnetic Permeability of a Material
Magnetic permeability (\(\mathrm{\mu}\)) is a measure of how easily a material becomes magnetized when exposed to an external magnetic field. It quantifies the ability of a material to support the formation of magnetic fields within itself.
Key Concepts:
Magnetic permeability relates the magnetic flux density \(\mathrm{\vec{B}}\) inside a material to the applied magnetic field strength \(\mathrm{\vec{H}}\):
\( \mathrm{\vec{B} = \mu \vec{H}} \)
Vacuum Permeability: Free space has a constant permeability, called the vacuum permeability (\(\mathrm{\mu_0}\)):
\( \mathrm{\mu_0 = 4\pi \times 10^{-7} \, H/m} \)
Relative Permeability: The permeability of a material is often expressed as:
\( \mathrm{\mu = \mu_r \mu_0} \)
where \(\mathrm{\mu_r}\) is the relative permeability of the material (dimensionless).
Variation with Material: Different materials respond differently to an external field:
- Diamagnetic: \(\mathrm{\mu_r < 1}\) (slightly less permeable than vacuum).
- Paramagnetic: \(\mathrm{\mu_r > 1}\) but close to 1.
- Ferromagnetic: \(\mathrm{\mu_r \gg 1}\), very high permeability due to domain alignment.
Dependence on Conditions: Unlike \(\mathrm{\mu_0}\), the permeability of a material is not constant. It varies with:
- Temperature (e.g., ferromagnetic materials lose magnetism above the Curie temperature).
- Orientation of dipoles within the material.
- Strength of the applied field (\(\mathrm{H}\)) — non-linear behavior in ferromagnetic materials.
Example :
A solenoid with a core material has \(\mathrm{\mu_r = 200}\). If the applied magnetic field strength is \(\mathrm{H = 50 \, A/m}\), calculate the magnetic flux density \(\mathrm{B}\).
▶️ Answer/Explanation
Step 1: Effective permeability of the core: \(\mathrm{\mu = \mu_r \mu_0 = (200)(4\pi \times 10^{-7}) \, H/m}\).
\(\mathrm{\mu \approx 2.51 \times 10^{-4} \, H/m}\).
Step 2: Use \(\mathrm{B = \mu H = (2.51 \times 10^{-4})(50)}\).
\(\mathrm{B \approx 1.26 \times 10^{-2} \, T}\).
Answer: The magnetic flux density is \(\mathrm{0.0126 \, T}\).