In this paper, we consider the Cauchy problem of the magnetic Zakharov system
in a cold plasma without the skin effect in two-dimensional space: \[
\begin{cases} & i E_{1t}+\Delta E_1-n E_1+\eta E_2
(E_1\overline{E_2}-\overline{E_1} E_2)=0, \\ & i E_{2t}+\Delta E_2-n E_2+\eta
E_1(\overline{E_1} E_2-E_1\overline{E_2})=0, \\ & n_t+\nabla \cdot
\textbf{v}=0, \\ & \textbf{v}_t+\nabla n+\nabla (|E_1|^2+|E_2|^2)=0, \\
\end{cases} \tag{G-Z} \] with initial data
$\left(E_{10}(x),E_{20}(x),n_{0}(x),\mathbf{v}_{0}(x)\right)$, where $\eta>0$
is a physical constant. Compared with the classical Zakharov system, (G-Z)
contains two extra cubic coupling terms
$E_2\left(E_1\overline{E_2}-\overline{E_1}E_2\right)$ and
$E_1\left(\overline{E_1}E_2-E_1\overline{E_2}\right)$ generated by the cold
magnetic field without the skin effect, which bring another difficulty. By
making a priori estimates corresponding for these extra terms, we obtain the
sharp lower bound of blow-up rate of the finite-time blow-up solutions to the
Cauchy problem of (G-Z) in the energy space
$\mathbb{H}_1=H^1(\mathbb{R}^2)\times H^1(\mathbb{R}^2)\times
L^2(\mathbb{R}^2)\times L^2(\mathbb{R}^2)$ provided the initial data satisfy
\begin{gather*} \frac{||Q||_{L^2(\mathbb{R}^2)}^2}{1+\eta}
<||E_{10}||_{L^2(\mathbb{R}^2)}^2+||E_{20}||_{L^2(\mathbb{R}^2)}^2
<\frac{||Q||_{L^2(\mathbb{R}^2)}^2}{\eta}, \end{gather*} where $Q$ is the
unique radially positive solution of the equation \begin{gather*} -\Delta
V+V=V^3. \end{gather*} Namely, there is a constant $c>0$ depending on the
initial data only such that for $t$ near $T$, \begin{gather*}
\left\|\left(E_1,E_2,n,\textbf{v}\right)\right\|_{H^1(\mathbb{R}^2)\times
H^1(\mathbb{R}^2)\times L^2(\mathbb{R}^2)\times
L^2(\mathbb{R}^2)}\geqslant\frac{c}{ T-t }, \end{gather*} where $T$ is the
blow-up time.
