$1.$ Consider two different metallic strips ($1$ and $2$) of the same material. Their lengths are the same, widths are $w_1$ and $w_2$ and thicknesses are $d_1$ and $d_2$, respectively. Two points $K$ and $M$ are symmetrically located on the opposite faces parallel to the $x$ - $y$ plane (see figure). $V _1$ and $V _2$ are the potential differences between $K$ and $M$ in strips $1$ and $2$ , respectively. Then, for a given current $I$ flowing through them in a given magnetic field strength $B$, the correct statement$(s)$ is(are)

$(A)$ If $w _1= w _2$ and $d _1=2 d _2$, then $V _2=2 V _1$

$(B)$ If $w_1=w_2$ and $d_1=2 d_2$, then $V_2=V_1$

$(C)$ If $w _1=2 w _2$ and $d _1= d _2$, then $V _2=2 V _1$

$(D)$ If $w _1=2 w _2$ and $d _1= d _2$, then $V _2= V _1$

$2.$ Consider two different metallic strips ($1$ and $2$) of same dimensions (lengths $\ell$, width w and thickness $d$ ) with carrier densities $n_1$ and $n_2$, respectively. Strip $1$ is placed in magnetic field $B_1$ and strip $2$ is placed in magnetic field $B_2$, both along positive $y$-directions. Then $V_1$ and $V_2$ are the potential differences developed between $K$ and $M$ in strips $1$ and $2$, respectively. Assuming that the current $I$ is the same for both the strips, the correct option$(s)$ is(are)

$(A)$ If $B_1=B_2$ and $n_1=2 n_2$, then $V_2=2 V_1$

$(B)$ If $B_1=B_2$ and $n_1=2 n_2$, then $V_2=V_1$

$(C)$ If $B _1=2 B _2$ and $n _1= n _2$, then $V _2=0.5 V _1$

$(D)$ If $B_1=2 B_2$ and $n_1=n_2$, then $V_2=V_1$

Give the answer question $1$ and $2.$ 

($A$) The voltmeter displays $-5 \mathrm{~V}$ as soon as the key is pressed, znd displays $+5 \mathrm{~V}$ after a long time

($B$) The voltmeter will display $0 \mathrm{~V}$ at time $t=\ln 2$ seconds

($C$) The current in the ammeter becomes $1 / e$ of the initial value after $1$ second

($D$) The current in the ammeter becomes zero after a long time