Resonance occurs widely in nature and is exploited in many man-made devices. Electric resonance is used in many circuits [1-5]; for example, radio and TV sets use resonance circuits to tune in to stations. In these devices, many frequencies reach the circuit simultaneously through the antenna, but significant current flow is induced only by frequencies at or near the circuit’s resonance frequency. By varying the inductance or capacitance, the device can be tuned to different stations. In physics, resonance is the tendency of a system to oscillate at a greater amplitude at the system’s resonance frequencies than at others. At these frequencies, even small periodic driving forces can produce large amplitude vibrations because the system stores the vibrational energy. In this work, we studied electromagnetic induction between two circular coils of wire: one is the source coil and the other is the pickup (or induction) coil, and report the characteristics of and control over the resonance and anti-resonance in the electromotive force (EMF) of electromagnetic induction through free space.
The experiment was performed based on Faraday’s law:
where 
and 
are the EMF induced in the induction coil and the magnetic flux passing through the induction coil, respectively [1,6]. The experimental setup consists of two circular coils of wire composed of an electrically conductive copper wire of cross-sectional radius 0.35 mm tightly wound into a series of loops of 5 - 320 turns, radius 7 cm and height 2 cm, as shown in
over time. Thus, an EMF is induced in the pickup coil. If the magnetic flux passing through the pickup coil is
, then the induced EMF is 
. As 
increases, 
increases, and the magnitude of 
is proportional to the rate at which the magnetic flux changes with time, so that faster changes give a stronger
. Here our question is: how does the magnitude of 
behave as 
increases, especially in the high frequency range of 10 kHz to a few MHz?
of the pickup coil and 2) the phase difference 
between the applied voltage (to the source coil) and the generated 
(in the pickup coil), respectively, as a function of the applied frequency 
(to the source coil). We expected that, according to Faraday’s law and the radiation resistance of the coil, 
would increase with increasing 
and then finally attain a new equilibrium, as shown by the black dashed line in 
of the pickup coil was 78˚ out-of-phase with respect to the wave applied to the source coil. As 
increased, the phase difference decreased until the phases matched, where the phase-matching means that the phase difference is less than 5˚. Meanwhile, the phase difference changed profoundly when the pickup coil became resonant. The 
of the pickup coil was 176˚ out-of-phase at frequencies between the resonance and anti-resonance peaks, that is, over a frequency range of 87 - 285 kHz.
at high frequencies. However, if the phase difference arose from the oscillations in the electric/magnetic dipoles or toroidal dipoles1 [8-14] induced by the electromagnetic radiation, the phase difference would be expected to depend on 
at high frequencies.
turns composed of an electrically conductive copper wire may be modeled as follows. For a coil of
, radius 7 cm and height 2 cm, as used here (see 
, where 
is the speed of light, for example,
; the former is on the order of
, the latter is on the order of
, where 
and 
are the angular frequency and the speed of light, respectively. The resonance frequencies obtained in our experiments were much lower than the resonance frequencies of the valence electrons or electric/magnetic dipoles [6,7]. Therefore, in the particular geometry studied here, the abrupt changes in the phase difference may have predominantly arisen from oscillations in the toroidal dipoles [13,14] in the induction coil.
and
could be adjusted, we examined the influence of three experimental parameters on the 
and
. Figures 3(a)-(c) show graphs of 
versus 
for various 1) voltages applied to the source coil (
= 2, 6, 10 and 14 V), 2) distances between the two coils (d = 10, 30, 100 and 1000 mm), and 3) numbers of turns of the pickup coil (
= 5, 50, 150, 250 and 320), respectively. Figures 4(a)-(c) show variations of 
and 
for
, 
and nB, respectively, obtained from 
was not influenced by 
or
, as shown by the black squares in Figures 4(a)-(b). The interval 
between 
and 
did not change significantly with increasing 
(
decreased dramatically as 
increased (
could be controlled by 
independently of
. On the other hand, as 
increased, 
and 
decreased together, as shown in 
and 
each show peculiar behavior. The 
and 
curves shown in 
does not depend on 
and 
could be described by an equation involving the logarithm of
. A plot of 
as a function of 
yielded a straight line, as indicated by the red line in the inset of 
can have the form “
”, where 
and 
are the proportionality constants and the vertical axis intercepts (or meaningless constants), re-
can have the form “
”, rather than “
”, and 
can also have the form “
”. The experimental results obtained for various combinations of
, 
and 
show that 
could be described by an equation of the form “
” with the condition
, and 
could be described by an equation of the form “
” with the condition
. It is noted that under the conditions of various 
turns, all the behaviors of 
and 
with 
were like the results shown in 
” might be negligible.
and 
for a given coil size. These results suggest the possibility of “a wireless power transfer station” that transmits power from a source coil to a large number of pickup coils