### Experimental analysis of minimum absorption bandwidth for Ag

Figure 2a, b shows the experimental results for the variation of absorption bandwidth for Ag NPs over the NPs’ size range of 9–44 nm, and the minimum absorption bandwidth (FWHM) occurs at the average NPs’ size of 22.4 nm.

The experimental and theoretical spectra for Ag (Fig. 2a, d) correlate well; they do not exhibit as much overlap of the absorption modes as in Au NPs (Fig. 1). This could be attributed to low damping effect on the SPR of Ag NPs, compared to the Au and Cu NPs (Ochoo et al. 2012). Of most significance here is that the turning point for Ag, like Au NPs, also falls within the anticipated NPs’ size range 10–30 nm, where the Au NPs’ sizes with the optimizing effect on both the cancer cells and the dye/solar cells fall.

### Calculated NPs’ sizes of minimum absorption bandwidth

Equation (4), derivable from Eq. (3) (Ochoo et al. 2012), is used here to confirm and validate the actual metal Au, Ag and Cu NPs’ sizes of the minimum absorption bandwidths.

$$\Delta \lambda = a\frac{{\gamma \lambda_{{\text{max}}}^{2} }}{2\pi c} + b\frac{{\lambda_{{\text{max}}}^{2} \pi kh}}{{4\pi R^{2} mc}},$$

(4)

where Δ*λ* is the absorption bandwidth of NPs, *λ*_{max} is wavelength of the peak location, *γ* is (bulk) damping constant term, *c* is speed of light, *h* is Planck’s constant, *R* is the radius of NPs, *m* is mass of electron, \(m = \left( {n_{x}^{2} + n_{y}^{2} + n_{z}^{2} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}}\), *c* is speed of light, ‘*a*’ and ‘*b*’ are constants (39.2 and − 25.4, respectively) obtained through the experimental data for gold from Bac et al. (2011).

The first term on the right-hand side is dependent on the damping term (*γ*) for the bulk material. The second term, however, is inversely dependent on the NPs’ size by the surface area equivalent of a nanosphere (4*πR*^{2}). Thus, the two terms should represent the right and left sides of the turning point of the curve, respectively (Figs. 1, 2b). In respect of this, at the turning point the condition for continuity is expected to be fulfilled by the two terms of Eq. 4. That is, the laws governing the plasmon resonance behavior on the right and left sides coincide at the turning point and should be equal. This would be true when Δ*λ* is set to zero. Thus, unless there are external damping factors on the NPs’ absorption bandwidth, the absorption should approximate to a spectral line (Δ*λ* = 0). With this condition, the spherical NPs’ sizes with minima absorption bandwidths would be given by Eq. 5.

$$R_{\text{o}} = \sqrt {\left( {\frac{{\left( { - b} \right)}}{{\left( { a } \right)}}\frac{\pi kh}{2m\gamma }} \right)} ,$$

(5)

where *R*_{o} is the radius of a spherical particle of the minimum absorption bandwidth.

From Eq. 5, *R*_{o} is independent of the wavelength of the absorbed light but is inversely dependent on the damping term (*γ*). To compute *R*_{o}, the experimental values for (*γ*), from Johnson and Christy (1972), have been used here. For gold *γ* is 0.072 eV (1.69 × 10^{13}/s); for Cu, *γ* is (2.31 × 10^{13}/s) and for Ag, *γ* is 0.021 eV (5.04 × 10^{12}/s). Because Au and Cu exhibit higher orders of SPR modes/damping (multipolar), we take *k* = 3 while for Ag *k* = 1 due to low modes (dipolar). The diameter size (2*R*_{o}) of the spherical Au NPs is found to be 21.7 nm, the same as from the experimental result of Link and El-Sayed (2000). For Cu, the calculated size is 19.7 nm and for Ag it is 24.2 nm. These values are in good agreement with the experimental values, therefore, confirming that the NPs’ sizes for the minimum absorption bandwidths lie within the 10–30 nm range. The corresponding approximate sizes in other shapes may be worked out based on the SA/VR equivalence. SA or SA/VR as the parameter for good performance would elicit the thought of SA-dependent phenomena such as the catalytic effect (chemical), heating effect (temperature) and electric field effect (charge density). The concern in this paper is about the phenomena that would justify the influence of the minimum absorption bandwidths on the metal NPs’ sizes of different shapes in the EMF therapy. In the photothermal effect theory, the limiting NPs’ size range for optimized results has been attributed to NPs uptake, distribution and retention by the cells (Choi et al. 2011). This perspective may lack scientific justification for the case of dyes and solar cells, whose enhanced light absorption optimizes at the same NPs’ sizes of Ag and Au as for the cancer cells (Photiphitak et al. 2010; Uppal et al. 2011). On the other hand, a catalytic effect would be expected to increase for the reducing NPs’ size (< 10 nm) because of the increasing SA/VR. However, unlike for the most effective NPs’ size (around 20 nm), they exhibit broader absorption bandwidths. It is the electric field (*E*) effect that appears to identify positively with the characteristics and parameters in question, especially the issue of the minimum absorption bandwidth.

### Electric field enhancement by metal nanoparticles

For the purpose of this paper, the electric field enhancement (*E*_{int}**/***E*_{o}) by metal NPs would be considered in the light of the parameters of Eq. 6, and the source of Eqs. (3, 4, 5) derived in a previous report (Ochoo et al. 2012). In comparison to Eq. 1, the expression of Eq. 6 includes the frequency of light, the permittivity of the medium and NPs, and nuclear charge (Ze), among other parameters. By Eq. 6, the electric field enhancement would be tunable and would optimize at resonance, when the denominator fulfills either condition *ω* = *ω*_{p} or *ε*_{p} = − 2*ε*_{m}.

$$\frac{{\varvec{E}_{{\text{int}}} }}{{\varvec{E}_{\text{o}} }} = \left( {\frac{{3\varepsilon_{\text{m}} }}{{\varepsilon_{\text{p}} + 2\varepsilon_{\text{m}} }}} \right)\left( {\frac{{\text{Ze}\rho }}{m}} \right)\left( {\frac{{\varepsilon_{\text{p}} - \varepsilon_{\text{m}} }}{{\left[ {\left( {\omega_{\text{p}}^{2} - \omega^{2} } \right)^{2} + \gamma^{2} \omega^{2} } \right]^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} }}} \right)$$

(6)

The resonance condition is easy to achieve by varying *ω* to match *ω*_{p}, Eq. 6 then reduces to Eq. 7. Thus, the electric field enhancement and its potential effect would depend on the metal type, the difference between the permittivity of the NPs and the medium (*ε*_{p} − *ε*_{m}) plus factors that may influence *ω*_{p} such as the NPs’ size, shape and the medium.

$$\frac{{\varvec{E}_{{\text{int}}} }}{{\varvec{E}_{\text{o}} }} = \left( {\frac{{3\varepsilon_{\text{m}} }}{{\varepsilon_{\text{p}} + 2\varepsilon_{\text{m}} }}} \right)\left( {\frac{{\text{Ze}\rho }}{m}} \right)\left( {\varepsilon_{\text{p}} - \varepsilon_{\text{m}} } \right)\left( {\frac{1}{{ \gamma \omega_{\text{p}} }}} \right)$$

(7)

Because of the negative sign in the term (ε_{p} − ε_{m}), the magnitude of the permittivity parameter (*ε*_{p}) may or may not impact on the magnitude of *E*_{int}. For the medium where *ε*_{p} ≪ *ε*_{m}, Eq. 7 approximates to Eq. 8 and *E*_{int} becomes independent of *ε*_{p}. Because cancer cells have higher water content than the normal cells, their measured dielectric constant (*ε*_{r}) values are about 64 (in vivo) and 62 (ex vivo) (Cho et al. 2006). Therefore, the approximation of Eq. 8 would fit the purpose of this paper. It predicts the amplification of the EMF’s electric field by a factor that would depend inversely on the SPR frequency (*ω*_{p}). The negative sign indicates that *E*_{int} would be opposite to the incident field *E*_{o}.

$$\varvec{E}_{{\text{int}}} = - \frac{{3\varepsilon_{\text{m}} }}{2}\frac{{\left( {\text{Ze}\rho } \right)}}{m}\frac{1}{{\gamma \omega_{\text{p}} }}\varvec{E}_{\text{o}}$$

(8)

From the Johnson and Christy (1972), the imaginary values for *ε*_{p} vary with the light wavelength, from about 2.0 in the absorption wavelength range of 520–985 nm for Au NPs to about 25.0 at the wavelength of 1939 nm. Because of the highly reduced frequency (*ω*_{p}) at the wavelength of 1939 nm, *E*_{int} would increase but not much (about 1.2 times) relative to region 520–985 nm. This suggests that the variation of *ε*_{p} does not influence the enhancement factor significantly. In a further analysis, we incorporate the influence of the spectral absorption bandwidth (Δ*λ*) by introducing it into Eqs. 7 or 8, through the parameter (*γ*). Here, we restrict ourselves to Eq. 8 for a while. Equation 9 gives the expression for (*γ*), from Eq. 4. Because the damping term (*γ*) is a property of the bulk material, it suffices to ignore the size-dependent term in Eq. 9 (second term). This leads to Eq. 10, with Δ*λ* and *λ*_{max} as the optical parameters which can be obtained from the absorption spectra of the metal NPs.

$$\gamma = \frac{2\pi c\Delta \lambda }{{a\lambda_{{\text{max}}}^{2} }} - \left( {\frac{b}{a}} \right)\frac{{2\pi^{2} kh}}{{4\pi R^{2} m}}$$

(9)

$$\varvec{E}_{{\text{int}}} = - \frac{{3\varepsilon_{\text{m}} }}{2}\frac{{\left( {\text{Ze}\rho } \right)}}{m}\frac{1}{{\omega_{p} }}\frac{{a \lambda_{{\text{max}}}^{2} }}{2\pi c\Delta \lambda }\varvec{E}_{\text{o}}$$

(10)

A parameter for nanosphere size (*R*) can be re-introduced through the de Broglie wave equation (*λ* = *h*/*μ*) and associated quantum expression for the kinetic energy of a free particle in a 3D box model, where the 1D energy model takes the form *E*_{k} = *n*^{2}*h*^{2}/8 m(2*R*)^{2}.

At resonance, let \(\omega = \omega_{\text{p}} = \frac{2\pi c}{{\lambda_{\text{p}} }}\) and set *λ*_{p} = *h*/*μ*; therefore,

$$\omega_{\text{p}} = 2\pi c\frac{\mu }{h} .$$

(11)

From the classical expression for kinetic energy of a free particle, \(E_{\text{k}} = \frac{1}{2}\mu^{2}\), hence, \(\omega_{\text{p}} = 2\pi c\frac{{\sqrt {2mE_{\text{k}} } }}{h}\), where \(E_{\text{k}} = \frac{{h^{2} }}{{8m\left( {2R} \right)^{2} }}\left( {n_{x}^{2} + n_{y}^{2} + n_{z}^{2} } \right)\) for a 3D box model. Hence,

$$\omega_{\text{p}} = \frac{k\pi c }{2R},\;\text{where}\;k = \left( {n_{x}^{2} + n_{y}^{2} + n_{z}^{2} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}}} .$$

(12)

Thus, Eqs. 10 and 12 can lead to Eq. 13.

$$\frac{{\varvec{E}_{{\text{int}}} }}{{\varvec{E}_{\text{O}} }} = - \frac{{3\varepsilon_{\text{m}} }}{2}\left( {\frac{{\text{Ze}\rho }}{m}} \right)\left( {\frac{a}{{\left( {2\pi c} \right) ^{2} }}} \right)\frac{{ (4R) \lambda_{{\text{max}}}^{2} }}{k \Delta \lambda }$$

(13)

Or, by setting \(2c = \omega \lambda_{{\text{max}}}\) at resonance (Eq. 11), Eq. 13 can be expressed as in Eq. 14.

$$\frac{{\varvec{E}_{{\text{int}}} }}{{\varvec{E}_{\text{O}} }} = - 3\varepsilon_{\text{m}} \left( {\frac{{\text{Ze}\rho }}{m}} \right) \frac{a}{{\omega^{2} }}\frac{{ \left( {2R} \right) }}{k \Delta \lambda }$$

(14)

By Eq. 14, for a given excitation light of frequency (*ω*), the electric field enhancement would be influenced by the ratio of the NPs’ size to the absorption bandwidth (2*R*/Δ*λ*), optimizing at the minimum value of Δ*λ* (Eq. 14). This would correspond to the turning point of the absorption bandwidth curves (Figs. 1 and 2b). As the NPs’ size (2*R*) or the excitation wavelength *λ*_{p} increases from left toward the turning point (Figs. 1 and 2b), Δ*λ* decreases then increases after the turning point. Thus, the enhancement of *E*_{int} is expected to decrease in either direction away from the turning point. Figure 3 shows the calculated electric field enhancement according to Eqs. 1 and 6, for the experimental Ag nanoparticles’ sizes 9–34 nm whose spectra are presented in Fig. 2a.

Figure 3 reveals that the optimized magnitude of *E*_{int} occurs at the average NPs’ size of 22.4 nm, which is the size at the minimum absorption bandwidth (turning point) in Fig. 2b. As the Ag NPs’ size shifts from 22.4 nm, the enhancement of *E*_{int} decreases, same trend as in the case of the influence of Au NPs’ size on the efficiency of EMF therapy (Zharov et al. 2003; Mackey et al. 2014). This suggests that the optimizing effect of the NPs’ size on the cancer cells is the same as for the electric field intensity around the NPs. Thus, the way the electric field enhancement follows the trend of the NPs’ size-dependent EMF effect on cancer cells could be suggestive of it having a major role in the EMF therapy. This suggestion would be considered in the context of power (P), to establish how the rate of energy delivered to NPs balances between the enhancement of electric field *E*_{int} and the dissipation as heat. From the report of El-Sayed et al. (2006), the laser energy (514 nm) kills malignant cells in the presence of Au NPs but not when there is no metal NPs, even at the source power densities four times higher (Giuliani and Soffritti 2010). This implies that the death of cancer cells is caused by the energy delivered to it through conversion by the metal NPs. Because there could be additional conditions, depending on the contributions of the electric field and the NPs’ sizes/shape but not featuring in the parameters of the model, a possible relation between *E*_{int} and the rate of energy delivery (power) would be discussed below. This is to be done through an alternative mathematical expression, which would bring together parameters of the electric field enhancement and those of the thermal effect. From it, we again find the electric field enhancement of NPs to be an important phenomenon and energy channel, determined by the NPs’ size, optical parameters and condition(s) for the EMF therapy of cancer cells.

### Rate of energy delivered through metal nanoparticles

After the alternating incident electric field of the EMF displaces the free electrons of charge (*q*) in the NPs, storing energy in them, an opposing restoring force associated with *E*_{int} would set in and dissipate some of the acquired energy during restoration. Depending on the frequency of the incident field and the intervening forces of the medium, the energy balance between the dissipated and that which remains in the system in terms of *E*_{int} would be dependent on the interacting forces. There are two important forces that would be relevant to this discussion. One is the damping force, it is dependent on the velocity of motion in a medium (*F*_{d}), and the other would be the restoring force due to the displacement of the electrons’ cloud from the core ion (*F*_{r}). For simplicity, *F*_{r} would be taken to be dependent on the displacement (*x*) and represented as *kx*. These two forces act in opposition to the driving force of the incident *E*_{o} and are bound to draw and convert the EMF energy into dissipative (thermal) and non-dissipative (electric field), respectively. This kind of interaction can be given by the general equation for damped motion (Eq. 15).

$$m\frac{{\text{d}^{2} x}}{{\text{d}t^{2} }} + kx + \beta \frac{{\text{d}x}}{{\text{d}t}} = F \,\text{Cos} \,\omega t$$

(15)

\(F_{\text{d}} = - \beta \frac{{\text{d}x}}{{\text{d}t}}\) (dissipative force) and \(F_{r} = - kx\) restoring non-dissipative force.If the displacement of the free electrons follows the driving force and expressed as *x* = *x*_{o}Cos *ωt*, then the restoring force would be *F*_{r} = − *kx*_{o}Cos *ωt* while the dissipative force *F*_{d} = *βωx*_{o}Sin *ωt*. Therefore, the rate of work done (power) by the restoring force (associated with the NPs-induced electric field) can be expressed by Eq. 16.

$$P_{\text{r}} = F_{\text{r}} \frac{{\text{d}x}}{{\text{d}t}} = - q\varvec{E}_{{\text{int}}} \frac{{\text{d}x}}{{\text{d}t}}$$

(16)

By introducing Eq. 14 into Eq. 16, we obtain Eq. 17, whose parameters would give the required clue about the factors to influence the system of power delivery by NPs. While the average power value over many cycles (〈*P*_{r}〉) would yield zero (Eq. 17), its root mean square value (〈*P*_{r}〉_{rms}) optimizes at the NPs’ size of minimum absorption bandwidth (Δ*λ*) (Eq. 18).

$$P_{\text{r}} = 3\varepsilon_{\text{m}} \left( {\frac{Ze\rho }{m}} \right) \frac{{\left( {2R} \right)a }}{ \omega \Delta \lambda k }q\varvec{E}_{\text{O}} \left( {x_{\text{o}}\, {\text{Sin }}\omega t} \right)$$

(17)

$$\left\langle {P_{\text{r}} } \right\rangle_{{\text{rms}}} = \frac{3}{2}\varepsilon_{\text{m}} \left( {\frac{Ze\rho }{m}} \right) \frac{{\left( {2R} \right)a }}{ \omega \Delta \lambda k }q\varvec{E}_{\text{O}} x_{\text{o}}$$

(18)

Since the displacement amplitude (*x*_{o}) of the electrons is proportional to the displacing field *E*_{o}, Eq. 18 can be expressed as Eq. 19.

$$\left\langle {P_{\text{r}} } \right\rangle_{{\text{rms}}} = \frac{3}{2}\varepsilon_{\text{m}} \left( {\frac{{\text{Ze}\rho }}{m}} \right) \frac{{\left( {2R} \right)a }}{ \omega \Delta \lambda k }q \left| {\varvec{E}_{\text{o}} } \right|^{2}$$

(19)

Similarly, the average power delivery to the dissipative force (heating) <*P*_{d}> would be given by Eq. 20, from Eq. 15. It is dependent directly on the square of the frequency

$$\left\langle {P_{\text{d}} } \right\rangle = \frac{1}{2}\beta \left( {\omega x_{\text{o}} } \right)^{2} = \frac{1}{2}\beta \omega^{2} \left| {\varvec{E}_{\text{o}} } \right|^{2}$$

(20)

Baffou et al. (2009) proposed the power of heat generation inside NPs by light absorption to be as in Eq. 21. It takes the same form as the corresponding expression for the dissipative force (Eq. 20). It suggests that the rate of EMF energy delivery to the NPs and, possibly, the rate of heating are directly dependent on the absorption cross-section of the NPs (*σ*_{abs}), the permittivity of the medium (*ε*_{m}) and intensity of the incident electric field.

$$\left\langle {P_{\text{d}} } \right\rangle = \frac{{nc\varepsilon_{\text{m}} }}{2}\sigma_{{\text{abs}}} \left| {\varvec{E}_{\text{o}} } \right|^{2} ,$$

(21)

where *n* is optical index of the surrounding medium.Assuming the energy delivered to the NPs is shared majorly between the electric field enhancement channel and the thermal conversion channel only, as per Eq. 15, then Eqs. 19 and 21 would represent their respective rates of energy delivery to the NPs and, possibly, to the cancer cells from the EMF. Thus, the overall energy transfer rate (EMF to NPs) would be given as in Eq. 22.

$$\left\langle P \right\rangle = \frac{3}{2}\varepsilon_{\text{m}} \left( {\frac{{\text{Ze}\rho }}{m}} \right) \frac{{\left( {2R} \right)a }}{ \omega \Delta \lambda k }q \left| {\varvec{E}_{\text{o}} } \right|^{2} + \frac{{nc\varepsilon_{\text{m}} }}{2}\sigma_{{\text{abs}}} \left| {\varvec{E}_{\text{o}} } \right|^{2}$$

(22)

Equation 22 suggests three important issues relevant for the understanding of the role of NPs’ size and their mechanisms of action. The first is that in the absence of the metal NPs such as Au or Ag, the benefit of the electric field channel (first term) would not be realized by the cancer cells. As a result, the EMF effect on cancer cells would be depending solely on the absorption cross-section of the cells (*σ*_{abs}), which would be very small. This may explain why, without metal NPs, even the higher energy powers of laser are not effective, suggesting that the thermal channel (heat) for the cells is of insignificant effect. Second, Eq. 22 suggests that the metal NPs do not act through the thermal channel term only, which would be to merely enhance the effective value of *σ*_{abs}. It introduces an additional channel (first term) as a likely complementing energy channel. Third, it suggests that it is because of the SPR in metal NPs the rate of energy extraction from EMF to the cancer cells is improved by the additional channel and the enhancement of *σ*_{abs}, depending on the magnitudes of the two terms in Eq. 22. If the optimizing effects of the NPs were to be associated with the heating effect only (yield and rate), as attributed to nanoshells and nanorods (Iancu 2013), then the NPs’ size effect would be influenced only by the *σ*_{abs}-dependent term (Eq. 22). On the other hand, if the EMF energy is shared between the two channels then the impact of the NPs size would be determined by the role and the magnitude of each of the energy channels in the EMF therapy. That is, both *σ*_{abs} and Δ*λ* would be involved. Since the absorption cross-section of spherical NPs is proportional to the physical surface area [4*πR*^{2} or 2*πR*(2*R*)], then it is proportional to the diameter of the particle. Therefore, the balancing ratio of the energy between the two channels (Eq. 22) would be dependent majorly on the variation of the ratio *σ*_{abs}:Δ*λ*, which can be approximated as in Eq. 23.

$$\sigma_{\text{abs}} : \Delta \lambda = \frac{{\sigma_{\text{abs }} }}{\Delta \lambda } \cong \frac{2R}{\Delta \lambda }$$

(23)

Thus, if the issue of the energy balancing and impact of NPs on the cancer cells is reduced to be the affairs of the diameter of NPs and the absorption bandwidth (2*R*/Δ*λ*), as earlier discussed in Eq. 14, the first term of Eq. 22 (electric field term) becomes very significant in the use of NPs. It is influenced inversely by the variation of Δ*λ*. This could explain why the inclusion of metal NPs in the EMF thermal therapy of cancer cells yields superior results compared to the unaided EMF (*σ*_{abs} only). It would also explain why the NPs-aided EMF effect optimizes as Δ*λ* reduces toward the minimum, which would explain the same trend seen for the NPs’ size-dependent enhancement of the light energy absorption by the dye/solar cells (Photiphitak et al. 2010). Based on the order of the magnitudes of the parameters in the first and the second terms of Eq. 22, for the spherical Au NPs of diameter 22 nm, we get the ratio for the second term to the first term (Eq. 22) to be about 2:5. This suggests that the electric field energy channel takes up an enormous amount of the EMF energy than the thermal channel during irradiation. In the EMF therapy, therefore, a higher electric field energy channel would be expected to induce higher and fast alternating force (impulse or shock) on the membrane and the ions on both sides of the cell membrane. This is likely to influence the redistribution of the ions on either side of the membrane, leading to change in the concentration gradients of these ions and the membrane potential (hyperpolarization). That is, because an electric field is accompanied by electric force whose magnitude is dependent on the charge type, it would act differently on the ions in its vicinity. The common charge carriers found on the cell membrane and in the cytoplasm are Ca^{2+}, Na^{+} and K^{+} (Pall 2013). The regions of a cell membrane with the divalent charges, Ca^{2+}, Mg^{2+} or Zn^{2+}, are likely to experience higher electric force than the monovalent ions (Na^{+} or K^{+}). This effect would ultimately alter the functioning of the cancer cells, where the division of cancer cells can be blocked (Lobikin et al. 2012; Persinger and Lafrenie 2014). Also, the penetration depth (range) of the induced alternating electric field (*E*_{int}), between the cell membrane and the cytoplasm, would depend on the frequency of the field (*E*_{int}). Equation 22 shows the electric field energy-dependent term (first term) to be dependent on the frequency (*ω*) of the EMF inversely. This suggests that while the electric field energy channel of the NPs would be more enhanced at very low frequencies, if other factors are constant, *E*_{int} and its penetration effect through the membrane would be low due to high capacitive impedance for low frequencies. On the other hand, very high-frequency fields encounter low impedance and would penetrate deeper through the membrane. However, by Eq. 22, very high frequency would lower the energy for the electric field channel (first term). Thus, although the electric field would deliver the higher frequency energy far into the cytoplasm, it would be low energy. This can explain the need for NPs’ sizes of absorption characteristics that can balance the energy intensity and range. Thus, the observed NPs’ size and absorption bandwidth effect on the efficacy of EMF therapy can be interpreted in the light of energy balancing, in which the electric field and thermal channels are complementary. A highly enhanced electric force (*F* = *μq**E*), where *μ* is an enhancement factor, can rupture the cancer cell membrane and cause increased permeability of the plasma membrane, and therefore, allowing into the cell the molecules to which the plasma membrane would be impermeable. This is likely to alter the behavior of the cancer cells and would lead to cell apoptosis. According to Pall (2013), the death of cancer cells can be initiated by the disruption of the plasma membrane, leading to influx of Ca^{2+} which causes cell damage. This is expected to be enhanced by transient effect of electric field as expressed in other studies (Choi et al. 2011). Also, the electric field is likely to cause change in the conformation of the proteins and enzymes embedded within the membrane, which is likely to depend on the strength and range of the field by the NPs’ size used. Elsewhere, laser radiations and pulsed electrical field have been reported to produce similar non-thermal effects on dipolar molecules, like enzymes, without increase in temperatures or dissipation of the energy (Amat et al. 2006). These are findings that agree well with Eq. 22, which suggest the existence of non-thermal-dependent energy channel (electric field). On the basis of the predictions of the proposed model and the existing evidence that the metal NPs-mediated EMF therapy of cancer cells trends with the absorption bandwidth, the enhanced electric field seems to have an important synergistic role to the thermal, as suggested by the model through the terms in Eq. 22.