Fundamentals of Physics Chapter 32 Solutions: Maxwell’s Equations; Magnetism of Matter
Halliday Resnick and Walker Volume 2 Solutions Chapter 32 ‘Maxwell’s Equations; Magnetism of Matter’ for JEE is a perfect learning guide for you. These talk about the magnetic field induced by a changing magnetic flux and are based on Maxwell’s equations and how it can be combined with Ampere’s law to give a single equation. In this chapter of Resnick Halliday Walker Solutions, You would learn the idea of fictitious or displacement current and how it applies to a capacitor which is being charged or discharged, the four fundamental laws of electromagnetism i.e. Gauss’ law of electricity and magnetism, Faraday’s law, and Ampere-Maxwell law. Also, you would identify lodestones and find out how the earth is a magnetic dipole, various laws around magnetism and electrons like Planck constant and Bohr magneton, diamagnetism, paramagnetism, and ferromagnetism.
Resnick Halliday and Walker Maxwell’s Equations; Magnetism of Matter includes 88 questions which are spread across 10 segments. You will get to hone your skills in calculating direction and magnitude of the electric field in a charging or discharging capacitor, find out displacement current, deduce magnetic dipole moment of an orbiting electron, and use Gauss’ law to find magnetic flux with the help of these exercise questions. The problems on a uniform and non-uniform magnetic flux, find the rate at which potential difference should be changed to cause a certain displacement current, magnetization of gas and many such questions would get you up to date with all kinds of questions which can come in Physics for JEE Mains and Advanced.
One needs to form a solid base in Physics for the level of difficult questions that can come in exams like JEE or NEET. The Halliday Resnick and Walker Fundamentals of Physics book is an exemplary source for managing these tough exams. You also need solutions to these questions in a way that clarifies the root concepts and makes even tough problems seem simple. Our team has strived to provide Solutions for Halliday Resnick and Walker Maxwell’s Equations; Magnetism of Matter in such a manner and you would find it a breeze to work them out with us.
Important Topics for Halliday Resnick and Walker Physics Volume 2 Solutions Chapter 32: Maxwell’s Equations; Magnetism of Matter
- Gauss’ Law for Electromagnetism: Every magnet is a dipole and monopoles do not exist in magnets. When we break a bar magnet with a north and a south pole, we form 2 magnetic dipoles as each fragment would have a north and South Pole. Hence it is concluded that a magnetic dipole is the simplest magnetic structure. Gauss law formally states this with its theory that the net magnetic flux Φx, through any Gaussian surface is 0.
- Extension of Ampere’s law by Maxwell –
Maxwell’s law states that when there is changing electric flux, a magnetic field is induced which is given by below formula:
- Ampere’s law gives the magnetic field generated by a current inside a closed-loop as
Where ienc is the total current enclosed by the loop.
- The above 2 laws can be given as a single equation as below:
- Displacement currents – When a current I is charging the plates of a capacitor, the electric field E between the plates changes. That changing field is termed as fictitious displacement current, id. This idea of displacement current helps in retaining the notion that there is continuous current in a capacitor, but this is not equivalent to the transfer of charge. The relationship can be defined as:
id = ε0 (dΦE/dt)
Where ΦE is magnetic flux due to the changing current
When we replace this in the Ampere-Maxwell equation we get:
Here id.enc is displacement current in the integration loop
The foundation of electromagnetism summarized below: The below table of Maxwell’s equations form the fundamental of electromagnetism:
|Gauss’ law for electricity||Depicts the relationship of net electric flux with the net enclosed electric charge|
|Gauss’ law for magnetism||Depicts the relationship of net magnetic flux with net enclosed magnetic charge|
|Faraday’s law||Depicts the relationship of the induced electric field due to changing magnetic flux|
|Ampere–Maxwell law||Depicts the relationship of the induced magnetic field due to changing electric flux and current|
- Earth as a magnetic dipole – Lodestones are naturally magnetized objects which were the first magnets ever discovered. Earth itself is a huge magnet and for points near the earth’s surface, the magnetic field is comparable to that of a huge bar magnet, a magnetic dipole that encompasses the centre of the planet. Its south pole is in the northern hemisphere. The direction of the magnetic dipole moment is at an angle of 11.5º to earth’s rotation axis.
- Spin magnetic dipole moment – The intrinsic angular momentum of an electron is called spin angular momentum or spin S. The spin magnetic dipole moment s associated with it is given by:
The minus sign denotes that the spin and magnetic moment are in opposite directions.
- Spin along z-axis – Spin cannot be measured on its own but its component along any axis can be determined. The spin Sz along the z-axis is given by:
Here, ms = spin magnetic quantum number = +- ½
H = Planck’s constant = 6.63 & 10-34 J*s
- Spin magnetic dipole moment along z-axis – Spin magnetic dipole moment s cannot be measured but its component along any axis can be measured given by:
- Orientation energy of spin magnetic dipole moment – When placed in an external magnetic field Mext, the orientation energy U associated with a spin magnetic dipole moment is given by:
- Orbital magnetic dipole moment – When an electron is in an atom, there is an additional angular momentum associated with it, called orbital angular momentum,
Lorb. The orbital magnetic dipole moment associated with its orbital angular momentum is given by:
Lorb can only be measured along an axis and is quantized. So its value along z-axis = ml h/2π
Here ml is orbital magnetic quantum number = 0, +-1, +-2, ….+-(limit)
“limit” refers to some largest allowed integer value for ml
Associate dipole moment along the z-axis is given by:
- Orientation energy of orbital magnetic dipole moment – When placed in an external magnetic field Mext, the orientation energy U associated with a spin magnetic dipole moment is given by:
- Diamagnetism – Materials that exhibit magnetism only when they are placed in an external magnetic field are diamagnetic materials. They form magnetic dipoles opposite to the external magnetic field. If the magnetic field is nonuniform then a diamagnetic material is repelled by the region of the higher magnetic field.
- Paramagnetism – Materials that have atoms with permanent magnetic dipole moment are paramagnetic materials. The orientations of the moments are random unless placed in an external field where they tend to align with the field. The extent to which the alignment happens within volume V is called magnetization M and given by:
M = measured magnetic moment/V
- When there is a complete alignment of all N atomic dipole moments it is called saturation and gives the maximum value of magnetization:
Mmax = Nμ/V
- Curie’s law states that the level of magnetization in a paramagnetic material is proportional to the magnitude of the external magnetic field Mext and inversely proportional to temperature T (in Kelvins):
Here C is Curie’s constant
- Ferromagnetism – In a Ferromagnetic material, once the dipole moments are aligned by an external magnetic field, they remain aligned partially in regions (domain) even after the external magnetic field is removed. This alignment gets removed at the Curie temperature of the material. If the field is nonuniform then a Ferromagnetic sample gets attracted to the region of the higher magnetic field.
Discussion of Exercises of Halliday Resnick and Walker Physics Volume 2 Solutions Chapter 32: Maxwell’s Equations; Magnetism of Matter
- The first exercise has 12 questions in which there are questions on capacitors where you need to calculate the direction of the electric or magnetic field, displacement current. There are sums on electrons and their spin orientation, ferromagnetic materials’ domain and rank them based on their degree of magnetization.
- The 2nd exercise has 4 questions on Gauss’ law of magnetic fields. The problems require you to calculate the magnitude and direction of magnetic flux in closed surfaces and objects like a die.
- The 3rd exercise is based on the concept of induced magnetic fields with a total of 8 questions. You would need to use Maxwell’s law of induction, Ampere’s law to solve these problems.
- The 4th set has 17 questions that test you extensively on displacement currents. These diverse questions on the potential difference of parallel plate capacitor, uniform and nonuniform displacement currents will give you the required expertise to solve any problem on this topic.
- The 5th set has 2 questions on magnets. The questions require you to calculate the magnitude and direction of magnetic flux in different parts of the earth’s surface.
- The 6th set is dedicated to magnetism and electrons with a total of 7 questions. Questions involve calculating orbital and spin angular momentum and dipole moment.
- The 7th set has 2 questions on diamagnetic samples where you have to find out the net magnetic dipole moment of an electron in an orbit.
- The 8th set is based on paramagnetism and has 7 questions. The questions involve Curie’s law to calculate the magnetization of materials like gas in a magnetic field.
- The 9th set has 8 questions on Ferromagnetism. Some of the highlights of the questions would be calculating needle rotation inertia of a compass in the earth’s magnetic field, the minimum energy required to turn a dipole, calculate current to be set up in a Rowland ring to get a certain magnitude of the toroidal field.
- The 10th set has 23 additional problems with questions touching most of the important topics of this chapter. There are questions on earth’s magnetism, the magnetic moment of the rotating charge, magnetic field as a function of time for parallel plate capacitor, the spin magnetic dipole moment of electrons.
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