When a potential difference V is applied between the ends of a wire of diameter d and length l , the current in the wire is I. What is the current when a potential difference of 2V is applied between the ends of a wire of the same material of diameter 2d and the length 2l ? Assume that the temperature of the wire remains constant.Â
resistance of wire:
"R = \\rho\\times\\frac{l}{S}"
where "\\rho" - electrical resistivity of material, l - length of wire, S - cross sectional area of wire.
current:
"I = \\frac{V}{R}"
then:
"R = \\frac{V}{I}"
this implies:
"\\frac{V}{I} = \\rho\\times\\frac{l}{S}"
Cross-sectional area for wire:
"S=\\pi \\times \\frac{d^2}{4}"
this implies:
"\\frac{V}{I} = \\rho\\times\\frac{4\\times l}{\\pi\\times d^2}"
expressing I:
"I=\\frac{V\\times \\pi \\times d^2}{4\\times \\rho \\times l}"
expressing current for second case:
"I_{2}=\\frac{2\\times V\\times \\pi \\times 4 \\times d^2}{4\\times \\rho \\times 2 \\times l}"
simplifying
"I_{2}=\\frac{V\\times \\pi \\times d^2}{\\rho \\times l}"
Comments
Leave a comment