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The dimensions of specific resistance ar...

The dimensions of specific resistance are

A

`[ML^(2)T^(-2)A^(-1)]`

B

`[ML^(3)T^(-3)A^(-2)]`

C

`[ML^(3)T^(-2)A^(-1)]`

D

`[ML^(2)T^(-2)A^(-2)]`

Text Solution

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The correct Answer is:
To find the dimensions of specific resistance (also known as resistivity), we start with the formula for resistance \( R \): \[ R = \frac{\rho L}{A} \] Where: - \( R \) is the resistance, - \( \rho \) is the resistivity (specific resistance), - \( L \) is the length of the conductor, - \( A \) is the cross-sectional area of the conductor. From this equation, we can express resistivity \( \rho \) as: \[ \rho = \frac{R \cdot A}{L} \] ### Step 1: Determine the dimensions of resistance \( R \) To find the dimensions of resistance, we can use the formula for heat generated in a resistor: \[ H = I^2 R T \] Where: - \( H \) is the heat generated, - \( I \) is the current, - \( T \) is the time. Rearranging gives us: \[ R = \frac{H}{I^2 T} \] ### Step 2: Find the dimensions of heat \( H \) The dimension of heat can be expressed as: \[ [H] = ML^2T^{-2} \] ### Step 3: Find the dimensions of current \( I \) The dimension of current \( I \) (in terms of ampere) is: \[ [I] = A \] ### Step 4: Find the dimensions of time \( T \) The dimension of time is: \[ [T] = T \] ### Step 5: Substitute the dimensions into the resistance formula Now substituting these dimensions into the resistance formula: \[ [R] = \frac{ML^2T^{-2}}{(A^2)(T)} = \frac{ML^2T^{-2}}{A^2 T} = \frac{ML^2}{A^2 T^3} \] Thus, the dimension of resistance \( R \) is: \[ [R] = ML^2T^{-3}A^{-2} \] ### Step 6: Substitute into the resistivity formula Now substituting the dimensions of \( R \) back into the resistivity formula: \[ \rho = \frac{R \cdot A}{L} \] Substituting the dimensions we have: \[ [\rho] = \frac{ML^2T^{-3}A^{-2} \cdot L^2}{L} = ML^3T^{-3}A^{-2} \] ### Final Result Thus, the dimensions of specific resistance (resistivity) are: \[ [\rho] = ML^3T^{-3}A^{-2} \]

To find the dimensions of specific resistance (also known as resistivity), we start with the formula for resistance \( R \): \[ R = \frac{\rho L}{A} \] Where: - \( R \) is the resistance, ...
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