30

In plain English, it is just Lenz’s law : Lenz's law, named after the physicist Emil Lenz who formulated it in 1834, states that the direction of the electric current which is induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes changes in the initial magnetic field. It is the basic ...


22

The minus sign is what makes Maxwell's equations obey causality, so it's a good thing it's there! To see this, you can write out the source-free Maxwell's equations with the sign of $\nabla \times \mathbf{E}$ reversed in Ampère's Law. If you then to follow the standard construction to extract the wave equation from Maxwell's equations, you would obtain ...


16

Duality is actually not $\mathbf{E}\leftrightarrow \mathbf{B}$ (I've used $c=1$), i.e. $(\mathbf{E},\,\mathbf{B})\to(\mathbf{B},\,\mathbf{E})$. It's $(\mathbf{E},\,\mathbf{B})\to(-\mathbf{B},\,\mathbf{E})$. Defining $\mathbf{F}:=\mathbf{E}+i\mathbf{B}$ is a popular way to check this; the above duality is $\mathbf{F}\to i\mathbf{F}$. It's instructive to ...


14

You are right, $\mathbf{B}$ is a vectorial quantity. Furthermore, it depends on position $\mathbf{r}$. Therefore we need to write $\mathbf{B}(\mathbf{r})$. The magnetic field around two current-carrying wires (with the currents flowing in the same direction) looks like this: (image from schoolphysics - electromagnetism - forces between currents) Notice ...


7

If the coil is superconducting the current is still there after you stop moving the magnet. In a normal-metal coil the current dies away due to the coil's resistance, and the extra energy is dissipated as heat.


7

Doesn't a battery do this? Also, capacitors. EDIT: With the edit, it looks like the premise of your question could be satisfied by a Van de Graff generator: https://en.wikipedia.org/wiki/Van_de_Graaff_generator which uses friction to strip electrons from a substance, and create an electrostatic potential.


7

Piezo electric cells convert mechanical energy to electric energy


7

It really comes from relativity, where one uses the field strength tensor: $$ F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}=\left(\begin{array}{cccc} 0 & E_x & E_y & E_z\\ -E_x & 0 & -B_z & B_x\\ -E_y & B_z & 0 & -B_y\\ -E_z & -B_y & B_x & 0\\ \end{array} \right)$$ When the indices are raised: $$F^{\mu\...


6

I'll just consider your first example; resolving this is sufficient to explain your other examples. Your error is that $\oint \vec{B} \cdot d\vec{r} \neq \vec{B} \oint d\vec{r}$. The field due to current $I_2$ is not constant over the Amperian loop you have drawn. However, the line integral of the magnetic field due to $I_2$ over the loop is zero (due to ...


6

So, according to Ampere's law, $$∮\mathbf{B}\cdot d\mathbf{l}=\mu_0I_1$$ Since, I've considered it to be an infinitely long wire, so $$B\oint dl=\mu_0I_1$$ This isn't a valid inference. When this inference is made in the textbook derivation of the B field around a long wire, we use the symmetry of system (the system is radially symmetric around the axis of ...


6

So we are only putting $mv^2/2$ energy which is gained by the wood but the extra output is current induced in the coil and magnet's motion, why is it not a violation to law of Conservation of energy? Actually, due to the magnetic radiation reaction force, the energy required in order to accelerate the magnet to $v$ is greater than $mv^2/2$. See https://en....


5

Lenz's Law is actually just the sign component of a more complex rule known as Faraday's Law of Induction (i.e. it says that the proportionality constant in Faraday's Law is negative). Because it is negative, changes in flux are opposed by any newly induced magnetic fields - meaning (among other things) that weakening fields get propped up and strengthening ...


5

No. The Maxwell equations describe how the electric and magnetic fields evolve given the charge density $\rho$ and current density $\mathbf J$. They do not describe the forces exerted on charged or magnetic material. For that, you need an additional input like the Lorentz force model for charged point particles, or a model for the force on a magnetic dipole.


5

What should really bother you, is not the minus sign in $(3)$. Is it's absence of in $(4)$!. The minus sign in $(3)$ actually prevents a run-away effect where an induced electric current would create a positive feedback on itself resulting in an unstable, ever-growing, electric current that: 1) would probably destroy your planet, and 2) violate energy ...


4

Ignatowski's equations follow from the representation of the fields by the vector and scalars as $$\begin{align}\mathbf{E}&=-\nabla \phi - \frac{\partial \mathbf{A}}{\partial t} \tag{1}\label{1}\\ \mathbf{B}&= \nabla \times\mathbf A \tag{2}\label{2} \end{align}$$ These two represent Faraday's induction law. When you substitute the integrals into $\...


4

In Faraday's equation: $$ E = -\frac{d\Phi}{dt} \tag{1} $$ the flux $\Phi$ is the total magnetic flux passing through the loop. That is, there is some magnetic field $B$ passing through the loop and we integrate this field across the area of the loop to get the flux $\Phi$. The magnetic field through the loop $B$ does not have to be constant across the loop. ...


4

When the magnet is brought closer to the the coil , magnetic field due to the magnet in the region of the coil changes with time and this gives rise to an electric field or an electric field is induced in that region . This induced electric field applies force on the electrons inside the coil and they start moving which gives rise to electric current and ...


4

There is an electrical component called a Möbius resistor that takes advantage of this weird geometry. The current flows in through the wire marked (+) and out through the wire marked (-). Because current flows in both directions around the ring, only a negligible magnetic field is generated. This can be important in high-power, high-frequency electrical ...


4

These fields do obey the superposition. The reason for this is that Maxwell equations are linear equations, and a sum of solutions of these equations is also a solution. In particular, plane waves, as well as the fields created by point charges, charge distributions, magnetic moments or electric currents are all solutions of Maxwell equations.


4

There's an approach from geometric algebra that considers the electromagnetic field as a field of bivectors, and with this geometric interpretation the electric and magnetic components can be rotated into each other. It's too long a discussion to attempt to explain this properly here; read the linked paper if you are interested in the details. I'll just give ...


3

The heat comes from current flowing through the wire, since we are told the wire has resistance. So it's just like your usual DC circuit with heat being dissipated in a resistor caused by charge carrier collisions with the molecules of the resistor. The energy comes from whatever is causing the varying magnetic field. This causes an induced EMF that induces ...


3

Yes, electromagnetic energy is stored in both E and B fields. The EM energy density is $U=\frac{E^2+B^2}{8\pi}$ in Gaussian units. Integrating U gives the energy.


3

Notice that the magnets are attached to the ends of a battery (the source of power). To experience a net force, the magnets must be in a non-uniform magnetic field. In this case the magnets conduct current, to the coil, which flows only between the magnets. A short segment of hollow magnetic torus produces a field which spreads rapidly at the ends and thus ...


3

This means that the rate of change of current keeps on decreasing as the time passes. It seems to me as the Inductor opposes the rate of change of current.However I am not sure. The inductor does oppose a rate of change in current. Consider the mechanical analogy of mass and inertia in which the inductance of an inductor is analogous to mass, and the ...


3

Good question. In my experience, most introductions to E&M don't give a great explanation of the exact similarities and differences between voltage and electromotive force (emf), and when you can and can't use the concepts interchangeably. Voltage and emf are both formally defined the exact same way, as the (negative) line integal of the electric field ...


3

The rate of doing work is $\vec{F}\cdot \vec{v}$. Since the magnetic component of the Lorentz force is $q\vec{v}\times \vec{B}$ then this force is always perpendicular to the velocity and does no work. In the absence of electric field from the other charges, the middle charge would execute a circular path at constant speed and kinetic energy. No work would ...


3

Mathematically you can surely see that $\vec{J}=0$ outside the wire and therefore the differential form of Ampere's law demands that $\nabla \times \vec{B} = 0$. I guess what is confusing is that curl is often described in terms of "curling field lines", but that isn't entirely accurate - straight field lines can have a curl and curved field lines ...


3

Since all calculations did by hft actually is right, we only proposed picture for this question how coil looks like from three different point.


3

(a) There will be no electromagnetic transfer of energy unless the ends of your coil are connected to a 'load' of some sort, or even just connected together so that there is a complete circuit. (b) There will be two pulses of voltage (emf), in opposite senses, one as the magnet enters the coil, and the other as it leaves. (c) To calculate the energy ...


3

There are no real perfect conductors, but if one were to exist, the electric field would indeed be zero. Currents would flow in the wire to cancel out the change in the magnetic field (the currents produce their own magnetic field which counteracts the change in the one you are producing).


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