Final answer:
If a wire's length is doubled and thickness is halved in a meter bridge setup, the new balance point will be 1/8 of the original balance length due to the increased resistance, which should correspond to option a. 1/8l.
Step-by-step explanation:
The meter bridge is an application of the Wheatstone bridge and is used for measuring resistance. When the wire of length 'l' is replaced with a wire of double the length and half the thickness, its resistance will change due to the properties of resistivity. The resistance (R) of a wire is directly proportional to the resistivity (p) and length (L), and inversely proportional to the cross-sectional area (A).
The cross-sectional area is proportional to the square of the diameter (or thickness, in this case). Therefore, if the length is doubled and the thickness is halved, the new resistance will be four times the original length (due to the length being doubled) and sixteen times greater due to the thickness being halved (as area is a square function). Therefore, the overall resistance will be eight times as much, so if the balance point was initially 'l', it will now be at 1/8l for the bridge to be balanced. This corresponds to option a. 1/8l.