$A$ metal wire of length $L$ and radius $r$ has a resistance $R$. If a wire of the same metal of length $2L$ and radius $3r$ is taken,then what will be its resistance?

$A$ cylindrical resistor is connected to a battery with emf $5 \ V$. The resistance per unit length varies as $\rho(x) = \rho_0 \left(\frac{x}{L}\right)^\alpha$,where $\rho_0$ and $\alpha$ are constants and $x$ is the distance from one end of the resistor. The product $\rho_0 L$ is $10 \ \Omega$,where $L$ is the length of the resistor. If the thermal power generated by the resistor is $20 \ W$,then the value of $\alpha$ is:

The resistance of a wire at $30^{\circ}C$ is $10\, \Omega$. To increase the resistance of the wire by $10\%$,the temperature should be .......... $^{\circ}C$. The temperature coefficient of resistance of the material of the wire is $0.002\, ^{\circ}C^{-1}$.

$A$ uniform wire of length $l$ and radius $r$ has a resistance of $100\, \Omega$. It is recast into a wire of radius $\frac{r}{2}$. The resistance of the new wire will be ............... $\Omega$.

Consider a wire of length $L$ with a resistance of $5 \Omega$. Applying an external force,the wire is elongated such that its length becomes $3 L$. Assuming the resistivity and density of the material remain unchanged,the resistance of the elongated wire is (in $\Omega$)

Two wires $A$ and $B$ are made up of the same material and have the same mass. Wire $A$ has a radius of $2.0 \ mm$ and wire $B$ has a radius of $4.0 \ mm$. The resistance of wire $B$ is $2 \ \Omega$. The resistance of wire $A$ is . . . . . . $\Omega$.