$\mathrm{d}_t W = \mathrm{d}_t \left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2\right) + \oint_{\partial V} (\mathbf{E} \wedge \mathbf{H}).\hat{\mathbf{n}} dS$$\mathrm{d}_t W = \mathrm{d}_t \int_V\left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2\right)\,\mathrm{d}\,V + \oint_{\partial V} (\mathbf{E} \wedge \mathbf{H}).\hat{\mathbf{n}} \,\mathrm{d}\,S$
$\mathrm{d}_t W = \mathrm{d}_t \left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2\right) + \oint_{\partial V} (\mathbf{E} \wedge \mathbf{H}).\hat{\mathbf{n}} dS$
$\mathrm{d}_t W = \mathrm{d}_t \int_V\left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2\right)\,\mathrm{d}\,V + \oint_{\partial V} (\mathbf{E} \wedge \mathbf{H}).\hat{\mathbf{n}} \,\mathrm{d}\,S$
The Lorentz force is the only force on a classical charged point particle (charge $q$), and - see Ben Crowell's answer about nonclassical particles with fundamental magnetic moment such as the electron). The magnetic component of the Lorentz force $q \mathbf{v} \wedge \mathbf{B}$, as you know, is always at right angles to the velocity $\mathbf{v}$, so there is no work done "directly" by a magnetic field $\mathbf{B}$ on this charged particle.
However, it is highly misleading to say that the magnetic field cannot do work at all - a time varying magnetic field always begets an electric field which does do work on a charge - you can't separate the electric and magnetic field from this standpoint. "Doing work" is about making a change on a system, and "drawing work from a system" is about letting the system change so that it can work on you. So we're always talking about a dynamic field in talking about energy transfer and in this situation you must think of the electromagnetic field as a unified whole. This is part of the meaning of the curl Maxwell equations (Faraday's and Ampère's laws).because:
- A time varying magnetic field always begets an electric field which can do work on a classical point charge - you can't separate the electric and magnetic field from this standpoint. "Doing work" is about making a change on a system, and "drawing work from a system" is about letting the system change so that it can work on you. So we're always talking about a dynamic field in talking about energy transfer and in this situation you must think of the electromagnetic field as a unified whole. This is part of the meaning of the curl Maxwell equations (Faraday's and Ampère's laws). Moreover, once things (i.e. charges and current elements) get moving, it becomes easier sometimes to think about forces from reference frames stationary with respect to them: Lorentz transformations then "mix" electric and magnetic fields in a fundamental way.
- A classical point charge belonging to a composite system (such as a "classical" electron in a metal lattice in a wire) acted on by the magnetic field through $q \mathbf{v} \wedge \mathbf{B}$ thrusts sideways on the wire (actually it shifts sideways a little until the charge imbalance arising from its displacement begets an electric field to support it in the lattice against the magnetic field's thrust). The magnetic field does not speed the charge up, so it does not work on the charge directly, but the sideways thrust imparted through the charge can do work on the surrounding lattice. Current elements not aligned to the magnetic field have torques on them through the same mechanism and these torques can do work. These mechanisms underly electric motors.
- Another way to summarise statements 1. and 2. is (as discussed in more detail below) that magnetic field has energy density $\frac{|\mathbf{B}|^2}{2\mu_0}$. To tap the energy in this field, you must let the magnetic field dwindle with time, and electric field arising from the time varying magnetic field can work on charges to retrieve the work stored in the magnetic field.
- The thinking of current elements shrunken down to infinitesimal sizes is a classical motivation for thinking about the interaction between magnetic fields and the nonclassical particles with fundamental magnetic moments, as in Ben Crowell's answer (I say a motivation because if you go too far classically with this one you have to think of electrons as spread out charges spinning so swiftly that their outsides would be at greater than light speed - an idea that put Wolfgang Pauli into quite a spin).
Another way toWe can put this statement is that magnetic field has energy density $\frac{|\mathbf{B}|^2}{2\mu_0}$. To tapmost of the energymechanisms discussed in this field, you must let the magnetic field dwindle with time,statements 1. and electric field arising from the time varying magnetic field can work on charges to retrieve the work stored in the magnetic field2.
To put this statement into symbols: suppose we wish to set up a system of currents ofof current density $\mathbf{J}$ in perfect conductors (so that there is no ohmic loss). Around the currentcurrents, there is a magnetic field; if we wish to increase the currentcurrents, we will cause a time variation in this magnetic field, whence an electric field $\mathbf{E}$ that pushes back on our currents. So in the dynamic period when our current changes, to keep the current increasing we must do work per unit volume on the currentcurrents at a rate of $\mathrm{d}_t w = -\mathbf{J} \cdot \mathbf{E}$.
The Lorentz force is the only force on a charged particle (charge $q$), and the magnetic component $q \mathbf{v} \wedge \mathbf{B}$, as you know, is always at right angles to the velocity $\mathbf{v}$, so there is no work done "directly" by a magnetic field $\mathbf{B}$.
However, it is highly misleading to say that the magnetic field cannot do work at all - a time varying magnetic field always begets an electric field which does do work on a charge - you can't separate the electric and magnetic field from this standpoint. "Doing work" is about making a change on a system, and "drawing work from a system" is about letting the system change so that it can work on you. So we're always talking about a dynamic field in talking about energy transfer and in this situation you must think of the electromagnetic field as a unified whole. This is part of the meaning of the curl Maxwell equations (Faraday's and Ampère's laws).
Another way to put this statement is that magnetic field has energy density $\frac{|\mathbf{B}|^2}{2\mu_0}$. To tap the energy in this field, you must let the magnetic field dwindle with time, and electric field arising from the time varying magnetic field can work on charges to retrieve the work stored in the magnetic field.
To put this statement into symbols: suppose we wish to set up a system of currents of current density $\mathbf{J}$ in perfect conductors (so that there is no ohmic loss). Around the current, there is a magnetic field; if we wish to increase the current, we will cause a time variation in this magnetic field, whence an electric field $\mathbf{E}$ that pushes back on our currents. So in the dynamic period when our current changes, to keep the current increasing we must do work per unit volume on the current at a rate of $\mathrm{d}_t w = -\mathbf{J} \cdot \mathbf{E}$.
The Lorentz force is the only force on a classical charged point particle (charge $q$ - see Ben Crowell's answer about nonclassical particles with fundamental magnetic moment such as the electron). The magnetic component of the Lorentz force $q \mathbf{v} \wedge \mathbf{B}$, as you know, is always at right angles to the velocity $\mathbf{v}$, so there is no work done "directly" by a magnetic field $\mathbf{B}$ on this charged particle.
However, it is highly misleading to say that the magnetic field cannot do work at all because:
- A time varying magnetic field always begets an electric field which can do work on a classical point charge - you can't separate the electric and magnetic field from this standpoint. "Doing work" is about making a change on a system, and "drawing work from a system" is about letting the system change so that it can work on you. So we're always talking about a dynamic field in talking about energy transfer and in this situation you must think of the electromagnetic field as a unified whole. This is part of the meaning of the curl Maxwell equations (Faraday's and Ampère's laws). Moreover, once things (i.e. charges and current elements) get moving, it becomes easier sometimes to think about forces from reference frames stationary with respect to them: Lorentz transformations then "mix" electric and magnetic fields in a fundamental way.
- A classical point charge belonging to a composite system (such as a "classical" electron in a metal lattice in a wire) acted on by the magnetic field through $q \mathbf{v} \wedge \mathbf{B}$ thrusts sideways on the wire (actually it shifts sideways a little until the charge imbalance arising from its displacement begets an electric field to support it in the lattice against the magnetic field's thrust). The magnetic field does not speed the charge up, so it does not work on the charge directly, but the sideways thrust imparted through the charge can do work on the surrounding lattice. Current elements not aligned to the magnetic field have torques on them through the same mechanism and these torques can do work. These mechanisms underly electric motors.
- Another way to summarise statements 1. and 2. is (as discussed in more detail below) that magnetic field has energy density $\frac{|\mathbf{B}|^2}{2\mu_0}$. To tap the energy in this field, you must let the magnetic field dwindle with time, and electric field arising from the time varying magnetic field can work on charges to retrieve the work stored in the magnetic field.
- The thinking of current elements shrunken down to infinitesimal sizes is a classical motivation for thinking about the interaction between magnetic fields and the nonclassical particles with fundamental magnetic moments, as in Ben Crowell's answer (I say a motivation because if you go too far classically with this one you have to think of electrons as spread out charges spinning so swiftly that their outsides would be at greater than light speed - an idea that put Wolfgang Pauli into quite a spin).
We can put most of the mechanisms discussed in statements 1. and 2. into symbols: suppose we wish to set up a system of currents of current density $\mathbf{J}$ in perfect conductors (so that there is no ohmic loss). Around the currents, there is a magnetic field; if we wish to increase the currents, we will cause a time variation in this magnetic field, whence an electric field $\mathbf{E}$ that pushes back on our currents. So in the dynamic period when our current changes, to keep the current increasing we must do work per unit volume on the currents at a rate of $\mathrm{d}_t w = -\mathbf{J} \cdot \mathbf{E}$.
The Lorentz force is the only force on a charged particle (charge $q$), and the magnetic component $q \mathbf{v} \wedge \mathbf{B}$, as you know, is always at right angles to the velocity $\mathbf{v}$, so there is no work done "directly" by a magnetic field $\mathbf{B}$.
However, it is highly misleading to say that the magnetic field cannot do work at all - a time varying magnetic field always begets an electric field which does do work on a charge - you can't separate the electric and magnetic field from this standpoint. "Doing work" is about making a change on a system, and "drawing work from a system" is about letting the system change so that it can work on you. So we're always talking about a dynamic field in talking about energy transfer and in this situation you must think of the electromagnetic field as a unified whole. This is part of the meaning of the curl Maxwell equations (Faraday's and Ampère's laws).
Another way to put this statement is that magnetic field has energy density $\frac{|\mathbf{B}|^2}{2\mu_0}$. To tap the energy in this field, you must let the magnetic field dwindle with time, and electric field arising from the time varying magnetic field can work on charges to retrieve the work stored in the magnetic field.
To put this statement into symbols: suppose we wish to set up a system of currents of current density $\mathbf{J}$ in perfect conductors (so that there is no ohmic loss). Around the current, there is a magnetic field; if we wish to increase the current, we will cause a time variation in this magnetic field, whence an electric field $\mathbf{E}$ that pushes back on our currents. So in the dynamic period when our current changes, to keep the current increasing we must do work per unit volume on the current at a rate of $\mathrm{d}_t w = -\mathbf{J} \cdot \mathbf{E}$.
However, we can rewrite our current system $\mathbf{J}$ with the help of Ampère's law:
$\mathrm{d}_t w = -\mathbf{J} \cdot \mathbf{E} = -(\nabla \wedge \mathbf{H}) \cdot \mathbf{E} + \epsilon_0 \mathbf{E} \cdot \partial_t \mathbf{E}$
then with the help of the standard identity $\nabla \cdot (\mathbf{E} \wedge \mathbf{H})=(\nabla \wedge \mathbf{E})\cdot\mathbf{H} - (\nabla \wedge \mathbf{H})\cdot\mathbf{E}$ we can write:
$\mathrm{d}_t w = -(\nabla \wedge \mathbf{E}) \cdot \mathbf{H} + \nabla \cdot (\mathbf{E} \wedge \mathbf{H})+\partial_t\left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2\right)$
and then with the help of Faraday's law:
$\mathrm{d}_t w = +\mu_0 \mathbf{H} \cdot \partial_t \mathbf{H} + \nabla \cdot (\mathbf{E} \wedge \mathbf{H})+\frac{1}{2}\epsilon_0 |\mathbf{E}|^2 = + \nabla \cdot (\mathbf{E} \wedge \mathbf{H})+ \partial_t\left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2\right)$
and lastly if we integrate this per volume expression over a volume $V$ that includes all of our system of currents:
$\mathrm{d}_t W = \mathrm{d}_t \left(\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2\right) + \oint_{\partial V} (\mathbf{E} \wedge \mathbf{H}).\hat{\mathbf{n}} dS$
(the volume integral becomes a surface integral by dint of the Gauss divergence theorem). For many fields, particularly quasi-static ones, as $V$ gets very big, the Poynting vector ($\mathbf{E} \wedge \mathbf{H}$ - which represents radiation), integrated over $\partial V$ is negligible, which leads us to the idea that the store of our work is the volume integral of $\frac{1}{2}\epsilon_0 |\mathbf{E}|^2+\frac{1}{2}\mu_0 |\mathbf{H}|^2$, so the magnetic field contributes to the stored work. It should be clear that this discussion is a general description of any dynamic electromagnetic situation and is wholly independent of the sign of $\mathrm{d}_t W$. So it applies equally whether we are working through the currents on the field or the field is working on us.
The above is very general: we can bring it into sharper focus with a specific example where it is almost wholly the magnetic field storing and doing work: say we have a sheet current circulating around in a solenoid shape so that there is a near-uniform magnetic field inside. For a solenoid of radius $r$, the flux through the solenoid is $\pi r^2 |\mathbf{B}|$ and the magnetic induction if the sheet current density is $\sigma$ amperes for each metre of solenoid is $|\mathbf{B}| = \mu_0 \sigma$. If we raise the current density, there is a back EMF (transient electric field) around the surface current which we must work against and the work done per unit length of the solenoid is:
$\mathrm{d}_t W = \sigma \pi r^2 \mathrm{d}_t |\mathbf{B}| = \frac{1}{2} \mu_0 \pi r^2 \mathrm{d}_t \sigma^2 = \pi r^2 \times \mathrm{d}_t \frac{|\mathbf{B}|^2}{2 \mu_0}$
This all assumes the rate of change is such that the wavelength is much, much larger than $r$. So now, the energy store is purely magnetic field: the electric field energy density $\frac{1}{2}\epsilon_0 |\mathbf{E}|^2$ is negligible for this example, as is the contribution from the Poynting vector (take the volume $V$ in the above argument to be a cylindrical surface just outside the solenoid: just outside the solenoid, the magnetic field vanishes and the Poynting vectors are radial at the ends of the cylinder so they don't contribute either. The above analysis works in reverse: if we let the currents run down, the electromagnetic field can work of the currents and thus the stored magnetic energy can be retrieved.