Electromagnetism Physics Problems: Get Expert Help with Plainmath

The Maxwell equation $\oint <!--\; \oint \; -->\stackrel{\to <!--\; \to \; -->}{E}.\stackrel{\to <!--\; \to \; -->}{dA}=\frac{Q_{in}}{{\mathrm{ϵ<!--\; ϵ\; -->}}_0}$ is also called
a. Faraday's Law of Electromagnetic induction
b. Gauss Law of Electricity
c. Ampere's Law
d. Gauss Law of Magnetism

Does it matter if I use AC or DC as a source to create an electromagnet?
For example, suppose I have a coil which has about 50 turns. If I use the same AC or DC voltage, which will create a stronger magnetic field?

Formula for Magnetic Force on a Current-Carrying Wire
There seems to be a pretty standard formula, that if a wire of length $\mathrm{\ell <!--\; \ell \; -->}$ carrying current $I$ is immersed in a magnetic field $B$, then the magnitude of the magnetic force is
$F_B=I\mathrm{\ell <!--\; \ell \; -->}Bsin\mathrm{\theta <!--\; \theta \; -->}$
, where the direction of $F_B$ is the direction of $\mathrm{\ell <!--\; \ell \; -->}\times <!--\; \times \; -->B$ (determined using right-hand rule).
But, these properties can all be summarized into the equation
$dB=\frac{{\mathrm{\mu <!--\; \mu \; -->}}_0}{4\mathrm{\pi <!--\; \pi \; -->}}\frac{Id\mathrm{\ell <!--\; \ell \; -->}\times <!--\; \times \; -->r}{r^2}$
I understand the first equation, but don't get the second one. How would I even read this equation or apply it? Do I have to integrate first? How would this work?
Thank you in advance for any help/clarifications.

From bar magnets to the magnetic field around a current carrying wire, the source of all magnetism is ________.a)electrical conductorsb)electrical insulators c)tiny pieces of irond)the orientation and motion of electronse)tiny domains of aligned protons

You have magnet, and you completely cover it with mylar. What happens to the magnetic field? Are the field lines altered? Is the strength of the magnetic field outside the mylar covering lessened?I

Imagine placing a compass at the middle of the bar magnet but on the opposite side of from the compasses shown in the image. In which direction would the compass needle point?

Assuming that we have a metamaterial that can slow down the velocity of propagation of changes in an EM field to around 15 000 m/s (without felling or applying any forces), can we build a propulsion system with two high frequency electromagnets with this material in between them? Because when the field interacts with the other electromagnet, the first is already off, not felling the reaction.

What happens when a bar magnet is cut in half?

Nowadays an unlimited amount of information can be transferred around the planet using wireless methods. Is that possible to transfer the information using natural electromagnetic field such as Earth magnetosphere? What else natural fields are able to carry clusters of information besides artificially generated waves?

Electric reactive power? Is it real or is the design accepted for calculation?

Drop a small bar magnet through a vertical plastic pipe, noting its speed of fall. Then do the same with a copper pipe. Whoa! Why the diference?

Explain why a bar magnet will attract an iron nail to either its north pole or its south pole, but attract another magnet to only one of its poles.

The plate capacitor is charged and discharged with sinusoidally changing electric current. Why does a capacitor emit electromagnetic radiation?

Magnetic force on a charged particle doesn't affect total energy
So there's this sentence in a book I am reading. It states "When a charged particle is placed within a magnetic field, despite the magnetic force acting on the particle, there will be no change in the total energy content of the particle. As the force acts along the radius, it is a non-effective force. So, no work is done by the particle. Hence, the change in total energy content is zero."
I'd really like to understand what this means. If someone could explain what they're trying to say here.
My first thought was that this is nothing complicated and a simple iteration of the force times component of displacement theorem. Since the magnetic field acts perpendicular to the charge the force also acts perpendicular to the electric field. So there's no resultant work.

A conducting wire is wrapped around an iron cylinder. Which statement is correct?
a) A current in the wire will turn the iron into a magnet.
b)Neither of other answers are correct.
c)If the iron is magnetized, a current will be produced in the wire.
d)Both of other answers are correct.

Assume you have an iron core in the interior of the solenoid. It is well known that the strength of the field should increase by a factor of several hundred inside the solenoid as a result of the iron core.
However, at the boundary between the iron core and the surrounding air, what happens to the magnetic field strength?

Force on a current carrying wire of arbitrary shape in a magnetic field
To obtain the net force by an external magnetic field on a current carrying wire, we divide the wire into small sections of infinitesimal lengths. The force due to the magnetic field on a given section is then given by $dF=i(dl\times <!--\; \times \; -->B)$, where B is the external magnetic field at that section.
Consider one of these sections. If the wire is not a straight line(has some random shape), then won't the current in a different section of the wire exert a force on the considered section due to the magnetic field produced due to that section?
I am aware that when we consider any closed system, we are not concerned with the internal forces operating within the system, as internal forces, always occurring in pairs, impart no effect to the system as a whole(to its centre of mass, to be specific).
However, this reasoning is only valid as long as the internal forces obey Newton's third law. But, as I have learnt, magnetic force due to charged particles DOES NOT obey Newton's third law. So, if I were to consider the current carrying wire as my system, the external magnetic field will indeed exert a force on the wire. But what about the internal forces and why are they not considered?
To summarize: WHY DOES A CURRENT CARRYING WIRE OF RANDOM SHAPE NOT EXERT A FORCE ON ITSELF?
Please correct me where I am wrong.

a) Define magnetic field.
b) A long straight wire carries 7 A current. At a distance 4 mm from the wire, a particle is moving parallel to it. If the charge of the particle is 8 \mu C and moves with 200 ms^{-1}. Find
1b) the magnetic flux density, 4 mm from the wire.
2b) the magnetic force exerted on the particle.

A particle with a charge of $5\mathrm{\mu <!--\; \mu \; -->}C$ is moving at $3\cdot <!--\; \cdot \; -->{10}^{6}m/s$ perpendicularly through a magnetic field with a strength of 0.06T. What is the magnitude of the force on the particle?

Magnetic force comparison
Two magnets of different size are tied on two sides of a wall with non-elastic rope on a horizontal surface and the opposite poles face each other. Why is the tension on both of the ropes the same? Shouldn't the tension be greater on the smaller magnet?