Hidden Momentum

Hidden momentum, field momentum, and electromagnetic impulse
David Babson and Stephen P. Reynolds
Department of Physics, North Carolina State University, Raleigh, North Carolina 27695

Robin Bjorkquist and David J. Griffiths
Department of Physics, Reed College, Portland, Oregon 97202

Received 1 February 2009; accepted 20 May 2009
Electromagnetic fields carry energy, momentum, and angularmomentum. The momentum density,
B , accounts among other things for the pressure of light. But even static fields can carry
0E
momentum, and this would appear to contradict a general theorem that the total momentum of a
closed system is zero if its center of energy is at rest. In such cases, there must be some other
nonelectromagnetic momenta that cancel the field momentum. What is the nature ofthis “hidden
momentum” and what happens to it when the electromagnetic fields are turned off? © 2009 American
Association of Physics Teachers.

DOI: 10.1119/1.3152712
I. INTRODUCTION
The linear momentum density carried by electromagnetic
fields is related to the Poynting vector1
℘em =

1
S=
c2

0

E

B.

1

The classic example is an electromagnetic wave see Fig. 1 .
When thewave strikes an absorber, its momentum is passed
along in the form of the pressure of light. But there are other
examples in which the fields are perfectly static, and yet the
electromagnetic momentum is not zero. Consider, for instance, the following configurations.
Capacitor in a magnetic field. A charged parallel-plate caˆ
pacitor with uniform electric field E = −Ey is placed in a
ˆ
uniformmagnetic field B = Bz, as shown in Fig. 2.2,3 Naively,
the electromagnetic momentum is4
ˆ
0EBAdx

pem = −

ˆ
= − BQdx ,

2

where A is the area of the plates, d is their separation, and Q
is the charge on the upper plate.
Magnetic dipole and electric charge. A magnetic dipole
ˆ
m = my is situated a distance a from a point charge q, as
shown in Fig. 3.5 The electromagnetic momentumis
pem =

0

4

1
qm
ˆ
x= 2 E
a2
c

m,

3

where E is the electric field at the location of the dipole.
Polarized magnetized sphere. A sphere of radius R carries
a uniform polarization P and a uniform magnetization M
see Fig. 4 .6 The momentum carried by the fields is
pem =

4
9

0R

3

M

P.

4

Coaxial cable. A long coaxial cable length l is connected
to abattery of voltage V at one end and a resistor R at the
other see Fig. 5 . The momentum carried by the fields is
2

pem =

lV
ˆ
x.
c 2R

5

It seems strange to say the least! for purely static fields
to carry momentum. Can this possibly be the whole story?
And what happens to the momentum when we turn off the
fields? In Sec. II we explore the latter question in what
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Am. J. Phys.77 9 , September 2009

http://aapt.org/ajp

would appear to be the simplest context: the parallel-plate
capacitor in a uniform magnetic field. We are led to a surprising paradox. In Sec. III we return to the first question “Is
this the whole story?” , to which the answer is no. Here we
develop the theory of “hidden momentum.” In Sec. IV we
work out the details for an electric dipole at thecenter of a
spinning, uniformly charged spherical shell, and resolve the
apparent paradox from Sec. II. In Sec. V we do the same for
an electric dipole inside a long solenoid. In Sec. VI we demonstrate that hidden momentum always cancels electromagnetic momentum, in the static case,7 and draw some general
conclusions about the nature of the hidden momentum.
II. CAPACITOR IN A UNIFORM MAGNETICFIELD
If the electric or magnetic field is turned off, the momentum originally stored in the fields must one would think be
converted into ordinary mechanical momentum. For example, in the case of the capacitor in a magnetic field, we
might connect a wire between the plates, allowing the capacitor to discharge slowly8 see Fig. 6 . According to the
Lorentz law, this wire will experience a force F...