Introduction
A general property of the dusty plasma system is the spontaneous self-excited oscillation of organized or random motion. This may lead to new instabilities in the presence of dust grains or influence the characteristics of plasma instabilities without dust. Weibel instability (WI) is one such electromagnetic instability which converts the kinetic energy of streaming electrons in plasma into magnetic energy capable of sustaining a collisionless shock. WI is important for an understanding of the energetic electromagnetic emissions of gamma-ray bursts and supernova explosions. WI mainly occurs due to the current neutralization of the beam-plasma interaction (Guskov, Reference Guskov2005; Velarde et al., Reference Velarde, Ogando, Eliezer, Martinez-Val, Perlado and Murakami2005; Zhou and He, Reference Zhou and He2007) and generates growing electromagnetic vibrations. WI for the first time was observed by Weibel (Reference Weibel1959) in 1959, taking bi-Maxwellian electron distribution function, and further a wide range of anisotropic plasma distributions had been studied. Fried (Reference Fried1959) gave a simple explanation of the instability as the superposition of many counter-streaming beams which resembles the two-stream instability.
Pokhotelov and Balikhin (Reference Pokhotelov and Balikhin2012) concluded that the frequency of the instability varies proportionally to the electron temperature anisotropy when observed in plasma with a non-zero external magnetic field. The WI in strongly magnetized microwave-produced plasma has been reported by Ghorbanalilu (Reference Ghorbanalilu2006). The effect of magnetic field on WI was examined by Ji-Wei and Wen-Bing (Reference Ji-Wei and Wen-Bing2005), where they have observed that the strong background of magnetic field stabilizes the WI in electron-ion plasmas. The study of space charge in the current filamentation has been conducted by Tzoufras et al. (Reference Tzoufras, Ren, Tsung, Tonge, Mori, Fiore, Fonseca and Silva2006), and the generation of the magnetic field was observed via the WI in interpenetrating plasma flows by Huntington et al. (Reference Huntington, Fiuza, Ross, Zylstra, Drake, Froula, Gregori, Kugland, Kuranz, Levy, Li, Meinecke, Morita, Petrasso, Plechaty, Remington, Ryutov, Sakawa, Spitkovsky, Takabe and Park2015). An experimental study of filamentation due to the WI in counter-streaming laser-ablated plasmas has also been conducted by Dong et al. (Reference Dong, Yuan, Gao, Liu, Chen, Jia, Hua, Qiao, Chen, Zhu, Zhu, Zhao, Ji, Sheng and Zhang2016). The important feature of WI is that it can make the ordinary waves unstable in the presence of temperature anisotropy, and this character of the instability has been investigated by many researchers in their work (Furth, Reference Furth1963; Hamasaki, Reference Hamasaki1968; Davidson, Reference Davidson, Rosenbluth and Sagdeev1983; Bashir and Murtaza, Reference Bashir and Murtaza2012; Ibscher and Schlickeiser, Reference Ibscher and Schlickeiser2014; Treumann and Baumjohann, Reference Treumann and Baumjohann2014).
Electromagnetic wave fluctuations are a subject of great attraction in the field of dusty plasma. Dahamni et al. (Reference Dahamni, Fentazi and Annou2005) have studied the excitation of electromagnetic waves via WI, in the presence of counter-streaming dust beams and agglomeration of dust grains. Dust kinetic Alfven and acoustic waves in a Lorentzian plasma (Rubab et al., Reference Rubab, Erkaev and Biernat2009), Kinetic Alfven wave instability in an unmagnetized dusty plasma (Rubab et al., Reference Rubab, Erkaev, Biernat and Langmayr2011) has been studied to study the effect of dust on these waves. Dispersion relation using many particle distribution functions for investigation on ordinary-wave WI in space plasmas has been derived by Rubab et al. (Reference Rubab, Chian and Jatenco-Pereira2016).
In this paper, we investigate the role of dust grains on electromagnetic wave modes generated via WI due to counter-propagating electrons. The instability analysis has been carried out in the following section. We have obtained the response of streaming electrons by fluid treatment and the expression for the growth rate of electromagnetic instability by first-order perturbation theory. The Results and Discussion and Conclusion are presented in the final two sections, respectively.
Instability analysis
We consider a dusty plasma comprising of two types of electrons that possess drift velocities 
${\rm \upsilon} _0\hat{z}$ and 
$-{\rm \upsilon} _0\hat{z}$, respectively, with ions and dust particles. The equilibrium density of ions is n i0, of dust grains is n d0, and that of electrons is n e0. The charge, mass, and temperature of the electrons, ions, and dust particles are denoted by (−e, m e, T e), (e, m i,T i), and (−Q d0, m d, T d), respectively. Initially, there is no external electric and magnetic field (E 0 = B 0 = 0). At equilibrium, the electrons acquire a thermal velocity v te = (T e/m e)1/2.
A dusty plasma is unstable to a magnetic field perturbation 
$B_1\hat{y}$ with a wave vector k along the x-axis. We assume that the electromagnetic perturbation due to the magnetic field 
$\vec{B}_1 = \hat{y}B_1\exp (i{\rm \omega} t-ikx)$ and the electric field associated with it is given by 
$\vec{E}_1 = \hat{z}\,E_1\exp (i{\rm \omega} t-ikx)$.
The electron's response to the above field can be given by the fluid equation of motion
$$\displaystyle{{{\rm \partial} {\rm \vec{\rm \upsilon}}} \over {{\rm \partial} t}} + ({\rm \vec{\rm \upsilon}} \cdot \nabla ){\rm \vec{\rm \upsilon}} = -\displaystyle{{e\vec{E}} \over {m_{\rm e}}}-\displaystyle{e \over {cm_{\rm e}}}{\rm \vec{\rm \upsilon}} \times \vec{B}$$and the equation of continuity
$$\displaystyle{{{\rm \partial} n} \over {{\rm \partial} t}} + \nabla \cdot (n{\rm \vec{\rm \upsilon}} ) = 0,$$
where n = n e0 + n e1exp(iωt − ikx) and 
${\rm \vec{\rm \upsilon}} = {\rm \upsilon} _0\hat{z} + {\rm \vec{\rm \upsilon}} _1\exp(i{\rm \omega} t-ikx)$.
In Eq. (1), the pressure term is neglected. Neglecting the pressure term can be justified as the phase velocity of the wave is greater than the electron thermal velocity
$\left( {{\rm i.e.},\, {{\rm \omega} / k} \gg v_{{\rm te}}} \right)$. On linearization, from Eqs (1) and (2), the perturbed density for electrons is obtained as follows:
$$n_{{\rm e}1} = \displaystyle{{-n_{{\rm e}0}k_x^2 eE_1{\rm \upsilon} _0} \over {im_{\rm e}{\rm \omega} ^3}},$$
where we have used 
$\vec{B}_1 = (c\vec{k} \times \vec{E}_1/{\rm \omega} )$. Perturbed dust grain density can also be obtained from Eq. (2) by replacing n e0 by n d0, e by −Q d0 (for negatively charged dust grains), υ0 by υd0, and m e by m d as 
$n_{{\rm d}1} = (n_{{\rm d}0}k_x^2 Q_{{\rm d}0}E_1{\rm \upsilon} _{{\rm d}0}/im_{\rm d}{\rm \omega} ^3)$.
But n d1 = 0 as υd0 = 0, that is, the dust grain number density will fluctuate only if they have an equilibrium velocity in plasma or the dust grains are mobile.
Whipple et al. (Reference Whipple, Northdrop and Mendis1985) and Jana et al. (Reference Jana, Sen and Kaw1993) have expressed the dust charge fluctuation in terms of an equation:
$$\displaystyle{{dQ_{{\rm d}1}} \over {dt}} + {\rm \eta} _1Q_{{\rm d}1} = -\vert {I_{{\rm e}0}} \vert \left( {-{\displaystyle{{n_{{\rm e}1}} \over {n_{{\rm e}0}}}}} \right),$$
where we have considered n i1 ~ 0 in the present analysis due to a high-frequency regime, η = |I e0|e/C g(1/T e + 1/T i − eϕg) (dust charging rate), I e0 is the equilibrium electron current collected by dust grains, Q d1 = (Q d − Q d0) (perturbed dust grain charge), 
$C_{\rm g} = a + a^2({\rm \lambda} _{{\rm De}}^{-1} )$ (dust grain's capacitance), a is the dust grain radius, and λDe is the electron Debye length. The expression for dust charge fluctuation in Eq. (4) can be rewritten as follows:
$$Q_{{\rm d}1} = {\displaystyle{{\vert {I_{{\rm e}0}} \vert } \over {i({\rm \omega} + i{\rm \eta} )}}}\left( {-{\displaystyle{{n_{{\rm e}1}} \over {n_{{\rm e}0}}}}} \right).$$Putting the value of n e1 from Eq. (3) in Eq. (5), we obtain the perturbed dust grain charge as follows:
$$Q_{{\rm d}1} = \displaystyle{{\vert {I_{{\rm e}0}} \vert } \over {({\rm \omega} + i{\rm \eta} )}}\displaystyle{{{\rm \upsilon} _0k_x^2 eE_1} \over {m_{\rm d}{\rm \omega} ^3}}.$$ Using Poisson's equation 
$\nabla \cdot \vec{E}_1 = 4{\rm \pi} n_{{\rm d}0}Q_{{\rm d}1}$, we obtain
$$\nabla \cdot \vec{E}_1 = {\rm \omega} _{{\rm pe}}^2 \displaystyle{{{\rm \beta} {\rm \upsilon} _0} \over {({\rm \omega} + i{\rm \eta} )}}\displaystyle{{k_x^2 E_1} \over {{\rm \omega} ^3}},$$
where ωpe = (4πn e0e 2/m e)1/2 is the electron plasma frequency, and β = (|I e0|/e)(n d0/n e0) is the dust plasma coupling parameter. Using the charge neutrality condition (Prakash and Sharma, Reference Prakash and Sharma2009), the dust plasma coupling parameter β can be written as 
$ {\rm \beta} = 0.397(1-(1/{\rm \delta} ))\,(a/v_{{\rm te}})(m_{\rm i}/m_{\rm e}){\rm \omega} _{{\rm pi}}^2,$ and 
${\rm \eta} = 10^{-2}{\rm \omega} _{{\rm pe}}\left( a/{{\rm \lambda}_{{\rm De}}} \right) (1/{\rm \delta})$, where η is the time scale of delay.
Using the charge neutrality condition, we obtain the following equation:
$$-en_{{\rm i}0} + en_{{\rm e}0} + Q_{{\rm d}0}n_{{\rm d}0} = 0 \; {\rm or}\; \displaystyle{{n_{{\rm d}0}} \over {n_{{\rm e}0}}} = ({\rm \delta} -1)\displaystyle{e \over {Q_{{\rm d}0}}},$$where δ = n i0/n e0 is the relative density of negatively charged dust grains in plasma.
The wave equation can be written using Maxwell's equations as follows:
$$\nabla ^2\vec{E}_1-\nabla (\nabla \cdot \vec{E}_1) + \displaystyle{{{\rm \omega} ^2} \over {c^2}}\vec{E}_1 = -\displaystyle{{4{\rm \pi} i{\rm \omega}} \over {c^2}}\vec{J}_{1z},$$where J 1z is the net perturbed current density.
For plasma electrons propagating in one direction, current densities are obtained by using Eq. (2) as
$$\vec{J}_{{\rm e}1x} = -n_0e^2E_1{\textstyle{{k_xv_0} \over {mi{\rm \omega} ^2}}}\;{\rm and}\;\vec{J}_{{\rm e}1z} = -n_0e^2E_1{\textstyle{1 \over {mi{\rm \omega}}}} \left[ {1 + {\textstyle{{k_x^2 v_0^2} \over {{\rm \omega}^2}}}} \right].$$Analogous expressions may be written for the perturbed current density of plasma electrons moving in other direction. Since the x-component of current density is proportional to υ0; therefore, they cancel each other in net current density. However, the z-component is proportional to υ02, thus giving the net perturbed current density as follows:
$$\vec{J}_1 = -2n_{{\rm e}0}e^2E_1\displaystyle{1 \over {im_{\rm e}{\rm \omega}}} \left[ {1 + \displaystyle{{k_x^2 {\rm \upsilon}_0^2} \over {{\rm \omega}^2}}} \right].$$ Putting the value of 
$\vec{J}_1$ from Eq. (10) and 
$\nabla \cdot \vec{E}_1$ from Eq. (7), we can rewrite Eq. (9) as follows:
$$\nabla ^2\vec{E}_1 + \nabla \left[ {{\rm \omega}_{{\rm pe}}^2 \displaystyle{{{\rm \beta} {\rm \upsilon}_0} \over {({\rm \omega} + i{\rm \eta} )}}\displaystyle{{k_x^2 E_1} \over {{\rm \omega}^3}}} \right] + \displaystyle{{{\rm \omega} ^2} \over {c^2}}\vec{E}_1 = -\displaystyle{{4{\rm \pi} i{\rm \omega}} \over {c^2}}\vec{J}_1.$$Further, solving Eq. (11), we obtain the following equation:
$${\rm \omega} ^4-(k_x^2 c^2 + 2{\rm \omega} _{{\rm pe}}^2 ){\rm \omega} ^2-2{\rm \omega} _{{\rm pe}}^2 k_x^2 {\rm \upsilon} _0^2 = \displaystyle{{i{\rm \beta}} \over {({\rm \omega} + i{\rm \eta})}}{\rm \omega} _{{\rm pe}}^2 \displaystyle{{k_x^3 {\rm \upsilon} _0{\rm c}^2} \over {\rm \omega}}. $$In the absence of dust δ(= n i0/n e0) = 1, that is, β → 0, then Eq. (12) transforms into
$$({\rm \omega} ^2-{\rm \omega} _ + ^2 )({\rm \omega} ^2-{\rm \omega} _-^2 ) = 0,$$where
$${\rm \omega} _ + ^2 = \lpar {k_x^2 c^2 + 2{\rm \omega}_{{\rm pe}}^2} \rpar + \displaystyle{{2{\rm \omega} _{{\rm pe}}^2 k_x^2 {\rm \upsilon} _0^2} \over {\lpar {k_x^2 c^2 + 2{\rm \omega}_{{\rm pe}}^2} \rpar }}$$and
$${\rm \omega} _-^2 = \displaystyle{{-2{\rm \omega} _{{\rm pe}}^2 k_x^2 {\rm \upsilon} _0^2} \over {\lpar {k_x^2 c^2 + 2{\rm \omega}_{{\rm pe}}^2} \rpar }}.$$Equation (14) of ω− corresponds to the usual dispersion relation of the WI. Now, we rewrite Eq. (12) in the presence of dust as
$$({\rm \omega} ^2-{\rm \omega} _ + ^2 )({\rm \omega} ^2-{\rm \omega} _-^2 )({\rm \omega} + i{\rm \eta} )\,{\rm \omega} = i{\rm \beta \omega} _{\rm pe}^2 k_x^3 {\rm \upsilon} _0c^2\cdot $$Further, we solve Eq. (13) under three limits:
Case I: Solving Eq. (13) for ω = ω+ + Δ1, we get the following equation:
$$\eqalign{\Delta _1 & = i{\rm \beta} \displaystyle{{{\rm \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {2{\rm \omega} _ + ({\rm \omega} _ + ^2 -{\rm \omega} _-^2 )({\rm \omega} _ + ^2 + {\rm \eta} ^2)}}\cr & + {\rm \beta \eta} \displaystyle{{{\rm \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {2{\rm \omega} _ + ^2 ({\rm \omega} _ + ^2 -{\rm \omega} _-^2 )({\rm \omega} _ + ^2 + {\rm \eta} ^2)}}.}$$where Δ1 is a small mismatch in the frequency due to dust particle.
Therefore, the growth rate corresponding to ω = ω+ + Δ1 is
$$\eqalign{\Gamma _1 = & {\mathop{\rm Im}\nolimits} (\Delta _1) \cr = &\displaystyle{{{\rm \beta \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {2{\rm \omega} _ + ({\rm \omega} _ + ^2 -{\rm \omega} _-^2 )({\rm \omega} _ + ^2 + {\rm \eta} ^2)}}\cdot} $$Case II: Again, solving Eq. (14) for ω = −iη + Δ2, we get the following equation:
$$\Delta _2 = \displaystyle{{-{\rm \beta \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {{\rm \eta} \,({\rm \omega} _ + ^2 + {\rm \eta} ^2)\,({\rm \omega} _-^2 + {\rm \eta} ^2)}}\cdot $$where Δ2 is the damping mode arising due to dust grain charge fluctuation.
Case III: Solving for ω = ω− + Δ3, we get the following equation:
$$\eqalign{\Delta _3 & = \displaystyle{{{\rm \beta \eta} {\rm \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {2{\rm \omega} _-^2 ({\rm \omega} _-^2 -{\rm \omega} _ + ^2 )({\rm \omega} _-^2 + {\rm \eta} ^2)}} \cr & + \displaystyle{{i{\rm \beta} {\rm \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {2{\rm \omega} _-({\rm \omega} _-^2 -{\rm \omega} _ + ^2 )({\rm \omega} _-^2 + {\rm \eta} ^2)}}.}$$where Δ3 is a small mismatch in the frequency due to dust particle.
Therefore, the growth rate corresponding to ω = ω− + Δ3 is
$$\eqalign{\Gamma _3 = &{\mathop{\rm Im}\nolimits} (\Delta _3) \cr = &\displaystyle{{{\rm \beta} {\rm \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2} \over {2{\rm \omega} _-({\rm \omega} _-^2 -{\rm \omega} _ + ^2 )({\rm \omega} _-^2 + {\rm \eta} ^2)}}\cdot} $$Equation (17) corresponds to the oscillations of WI. Further solving Eq. (17) in the limit of the negligible decay rate of dust charge fluctuations, that is η ≈ 0, and using Eqs (13) and (14), we obtain the frequency of WI oscillation as
$$\displaystyle{{{\rm \beta} k_xc^3} \over {4\sqrt 2 {\rm \omega} _{{\rm pe}}{\rm \upsilon} _0^2}}. $$The usual expression of WI ( Chen, Reference Chen2006), which is obtained from Eq. (14), is
$${\rm \gamma} = \displaystyle{{\sqrt 2 {\rm \omega} _{{\rm pe}}({{{\rm \upsilon}_0} / c})} \over {{[{1 + {{2{\rm \omega}_{{\rm pe}}^2} / {k_x^2 c^2}}}]}^{1/2}}}.$$This growth rate approaches saturation on electron cyclotron frequency and is set to oscillation [cf. Eq. (20)] due to the presence of dust grains in plasma with a frequency proportional to dust coupling parameter β and inversely proportional to electron streaming velocity. The electric and magnetic fields associated with the instability are 90° out of phase, and the growth of magnetic field results in the filamentation structure of WI.
Results and discussion
The dusty plasma parameters used for the calculations are: ion plasma density ni0 = 108 cm−3, electron plasma density n e0 = 0.1 × 108−1 × 108 cm−3, mass of dust grains m d = 1012 mp (for 1 µm grain assuming a mass density of ~1 g/cm3), temperature of ions and electrons T i ≈ T e ≈ 0.2 eV, dust density n d0 = 1 × 104 cm−3, m i/m e = 7.16 × 104 (potassium), and the average size of the dust grain a = 2 µm. Using Eq. (19), we have plotted the growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function of perpendicular wave number k x(cm−1) for different values of 
${\rm \delta} = {{n_{{\rm i}0}} /{n_{{\rm e}0}}}$ (relative density of negatively charged dust grains) (cf. Fig. 1). It is observed that the WI grows in the direction of increasing perpendicular wave number. In the present work, we have considered two types of electrons that possess drift velocities 
${\rm \upsilon} _{0}\hat{z}$ and 
$-{\rm \upsilon} _{0}\hat{z}\,.$ The perturbed magnetic field exerts a force 
$(-{\textstyle{e \over c}}{\rm \vec{\rm \upsilon}} _0 \times B_1)$ (along + x-axis) on the first type of electrons and force 
$( + {\textstyle{e \over c}}{\rm \vec{\rm \upsilon}} _0 \times B_1)$ (along – x-axis) on the second type of electrons. The electrons acquire a velocity in the x and −x directions, respectively. Since B 1 is a function of x, this velocity has finite divergence 
$(\nabla \cdot {\rm \upsilon} _1\ne 0)$ and gives rise to density perturbation. The density perturbations of the two types of electrons are out of phase by 180°. They couple with 
${\rm \upsilon} _0\hat{z}$ and 
$-{\rm \upsilon} _0\hat{z},$ respectively, to produce a current
$\vec{J}_1(1) = -n_1e{\rm \vec{\rm \upsilon}} _0 + (-n_0e{\rm \vec{\rm \upsilon}} _1)$ due to the first type of electrons and 
$\vec{J}_1(2) = n_1e{\rm \vec{\rm \upsilon}} _0-n_oe{\rm \vec{\rm \upsilon}} _1$ due to the second type of electrons in the z-direction. Total current density in plasma is given by Eq. (10). This current produces the magnetic field in the y-direction, enhancing original magnetic field perturbation. Thus, the perturbation grows with time at the expense of kinetic energy of the counter-streaming electrons. The density of dusty plasma in this scenario of counter-streaming electrons rises due to the combined effect of electron two-stream instability and electron-capturing tendency of dust; hence, electromagnetic wave generated due to WI becomes unstable and grows which is also observed by Ross et al. (Reference Ross, Park, Berger, Divol, Kugland, Rozmus, Ryutov and Glenzer2013) in which they have taken counter-streaming plasma flows. In our case, the growth rate with a perpendicular wave number in the presence of dust resembles with one observed by Huntington et al. (Reference Huntington, Fiuza, Ross, Zylstra, Drake, Froula, Gregori, Kugland, Kuranz, Levy, Li, Meinecke, Morita, Petrasso, Plechaty, Remington, Ryutov, Sakawa, Spitkovsky, Takabe and Park2015) in the absence of dust.
Fig. 1. Growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function of perpendicular wave number k x (cm−1) for δ = 2, 3, 4, and 5.
Again, using Eq. (19), Figure 2 is plotted which shows the variation of growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as the function of the parallel wave number k z(cm−1) for different values of δ = n i0/n e0. It is inferred from Figure 2 that the growth rate of unstable mode decreases as the parallel wave number increases. This shows that the growth rate of instability decreases in the direction of self-generated magnetic field (Huntington et al., Reference Huntington, Fiuza, Ross, Zylstra, Drake, Froula, Gregori, Kugland, Kuranz, Levy, Li, Meinecke, Morita, Petrasso, Plechaty, Remington, Ryutov, Sakawa, Spitkovsky, Takabe and Park2015) because dust and ion in the plasma get excited in the direction of magnetic field but being heavy and stationary they oscillate at their places, hence reducing the energy of electromagnetic plasma wave along the direction of counter-streaming electrons or self-generated magnetic field. This decrease in the growth rate of unstable mode in the direction of parallel wave number, and an increase in the direction of the perpendicular wave number is also reported by Lazar et al. (Reference Lazar, Schlickeiser, Poedts and Tautz2008).
Fig. 2. Growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function of parallel wave number k z (cm−1) for δ = 2, 3, 4, and 5.
Figure 3 shows the growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function of dust grain size a (cm) for different values of δ keeping all the other parameters [cf. Eq. (19)] the same as used for plotting Figure 1. The modification in the growth rate of instability is due to the presence of a large number of dust particles, as the electron-capturing tendency of these particles and dust charging process (Barkan et al., Reference Barkan, D'angelo and Merlino1994) change the rate of energy transfer between plasma wave and dust particles. This effect of dust grains on the ambient plasma wave has been observed by many researchers in their work. It is observed in Figure 3 that the growth rate of unstable mode first increases, and after acquiring the highest value, it becomes constant for all the values of δ. The growth rate value increases because on adding the dust grains in ambient plasma or by increasing the size of dust grains, the freely moving counter-streaming electrons approach them, raising the surface potential of particles of dust, as a result average dust grain charge Q d0 also increases which helps the instability to grow; hence, the enhancement in its growth rate is observed. This result is qualitatively similar to the results of Sharma and Sugawa (Reference Sharma and Sugawa1999), Prakash et al. (Reference Prakash, Sharma, Vijayshri and Gupta2013), and Prakash et al. (Reference Prakash, Sharma, Vijayshri and Gupta2014). The transfer of energy from counter-streaming electrons accelerated by a self-generated magnetic field to an electromagnetic wave via dust particles modifies the growth rate of instability. It is also observed in Figure 3 that when the size of dust grain becomes greater than 1.5 × 10−4 cm, the growth rate becomes almost constant for all the values of δ. The reason for this is that on further increasing the size of dust grains, the average dust grain charge leads to saturation, as dust grain grabbed enough number of electrons. Also, the plasma system becomes stable because strong self-generated magnetic field overpowers the growth rate (Rubab et al., Reference Rubab, Chian and Jatenco-Pereira2016).
Fig. 3. Growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function of the size of dust grains a (cm) for δ = 2, 3, 4, and 5 with k x = 0.5.
Using Eq. (19), we have plotted the growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ with respect to perpendicular wave number for different values of the velocity of electrons (υ0) say 
$2 \times 10^7$, 
$4 \times 10^7$, 
$6 \times 10^7$, and 
$8 \times 10^7\,{\rm cm}/{\rm s}$, taking δ = 3 and dust grain size a = 1.5 × 10−4 cm (cf. Fig. 4). It is observed that the growth decreases first decreases and after acquiring the lowest value it starts increasing gradually and goes to its maxima for all the values of υ0. As the velocity of electrons increases, the efficiency of instability to convert the kinetic energy of the system to magnetic energy takes the energy of electrons to least and the growth rate of plasma wave becomes minimum. The magnetization of dusty plasma due to WI via counter-streaming electrons sharply increases the acceleration and hence the energy of electrons which stimulate them and then increase the growth rate.
Fig. 4. Growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function of perpendicular wave number kx (cm−1) for 
${\rm \upsilon} _0 = 2 \times 10^7$, 
$4 \times 10^7$, 
$6 \times 10^7$, and 
$8 \times 10^7\,{\rm cm}/{\rm s}$, for δ = 3 and dust grain size a = 1.5 × 10−4 cm.
Moreover, using the same Eq. (19), the variation in the growth rate of instability 
$\Gamma _3\,({\rm s} ^{-1})$ with respect to δ (δ = 1 is for without dust as β = 0) by taking 
${\rm \upsilon} _0 = 4 \times 10^7\,{\rm cm}/{\rm s}$ and a = 1.5 × 10−4 cm is plotted as Figure 5, and we found that it is increasing with a relative density of negatively charged dust particles (δ). Our results are qualitatively similar to the work done in the field of dusty plasma (Barkan et al., Reference Barkan, D'angelo and Merlino1994; Chow and Rosenberg, Reference Chow and Rosenberg1995; Sharma and Sugawa, Reference Sharma and Sugawa1999; Sharma et al., Reference Sharma, Sharma and Walia2012; Prakash et al., Reference Prakash, Sharma, Vijayshri and Gupta2014). This is because of the electrons shielding nature of dust grains that helps in increasing the capacity of counter-streaming electrons to transfer their kinetic energy acquired by magnetization of dusty plasma to WI.
Fig. 5. Growth rate 
$\Gamma _3\,({\rm s} ^{-1})$ as a function δ for 
${\rm \upsilon} _0 = 4 \times 10^7\,{\rm cm}/{\rm s}$ and dust grain size a = 1.5 × 10−4 cm.
Conclusion
The counter-propagating electrons in an unmagnetized dusty plasma have the capability of generating electromagnetic waves via WI. In the present work, along with the growth rate of instability, a damping mode with frequency 
$ (-{\rm \beta} {\rm \omega} _{{\rm pe}}^2 k_x^3 {\rm \upsilon} _0c^2)/ ({\rm \eta} \,({\rm \omega} _ + ^2 + {\rm \eta} ^2)\,({\rm \omega} _-^2 + {\rm \eta} ^2))$ is observed, which is mainly due to a well-known dust charge process in the dust plasma system. It is found that the growth rate of unstable mode increases with dust grains’ size and with increasing perpendicular wave number; however, it saturates for higher values of dust grain size. The negatively charged dust grains contribute to enhancing the growth rate when observed for different velocities of streaming electrons. Our work may be beneficial in studying the effect of dust particles in the magnetotail region of planetary magnetospheres (Lui et al., Reference Lui, Yoon, Mok and Ryu2008).
