Units Dimensions And Measurements

Step 1: Dimensional formula for EMF (Electromotive Force)

%5Ctext%7BEMF%7D%20%3D%20%5Cfrac%7B%5Ctext%7BWork%7D%7D%7B%5Ctext%7BCharge%7D%7D%20%3D%20%5Cfrac%7BML%5E2T%5E%7B-2%7D%7D%7BAT%7D%20%3D%20ML%5E2T%5E%7B-3%7DA%5E%7B-1%7D

So,

S%20%3D%20%5Cfrac%7B%5Ctext%7BEMF%7D%7D%7B%5Ctext%7Btime%7D%7D%20%3D%20%5Cfrac%7BML%5E2T%5E%7B-3%7DA%5E%7B-1%7D%7D%7BT%7D%20%3D%20ML%5E2T%5E%7B-4%7DA%5E%7B-1%7D%0A%5CRightarrow%20S%5E2%20%3D%20M%5E2L%5E4T%5E%7B-8%7DA%5E%7B-2%7D

Step 2: Dimensional formula for electric conductivity $%5Csigma$

%5Csigma%20%3D%20M%5E%7B-1%7DL%5E%7B-3%7DT%5E3A%5E2

Step 3: Dimensional formula for thermal conductivity $k$

k%20%3D%20%5Cfrac%7B%5Ctext%7BEnergy%7D%7D%7B%5Ctext%7BTime%7D%20%5Ccdot%20%5Ctext%7BLength%7D%20%5Ccdot%20%5Ctext%7BTemperature%20difference%7D%7D%20%3D%20%5Cfrac%7BML%5E2T%5E%7B-2%7D%7D%7BT%20%5Ccdot%20L%20%5Ccdot%20K%7D%20%3D%20MLT%5E%7B-3%7DK%5E%7B-1%7D

Step 4: Combine to get $%5Calpha$

%5Calpha%20%3D%20%5Cfrac%7BS%5E2%20%5Ccdot%20%5Csigma%7D%7Bk%7D%20%3D%20%5Cfrac%7BM%5E2L%5E4T%5E%7B-8%7DA%5E%7B-2%7D%20%5Ccdot%20M%5E%7B-1%7DL%5E%7B-3%7DT%5E3A%5E2%7D%7BMLT%5E%7B-3%7DK%5E%7B-1%7D%7D

Simplify: - Mass:

M%5E%7B2%20-%201%20-%201%7D%20%3D%20M%5E0

- Length:

L%5E%7B4%20-%203%20-%201%7D%20%3D%20L%5E0

- Time:

T%5E%7B-8%20%2B%203%20%2B%203%7D%20%3D%20T%5E%7B-2%7D

- Current:

A%5E%7B-2%20%2B%202%7D%20%3D%20A%5E0

- Temperature:

K%5E1

%5CRightarrow%20%5Cboxed%7B%5BM%5E0L%5E0T%5E%7B-2%7DK%5E1%5D%7D

Wait — looks like a miscalculation. Let's double-check final step: Actually: Numerator:

S%5E2%20%5Ccdot%20%5Csigma%20%3D%20M%5E2L%5E4T%5E%7B-8%7DA%5E%7B-2%7D%20%5Ccdot%20M%5E%7B-1%7DL%5E%7B-3%7DT%5E3A%5E2%20%3D%20ML%5E1T%5E%7B-5%7D

Denominator:

k%20%3D%20M%5E1L%5E1T%5E%7B-3%7DK%5E%7B-1%7D

Now:

%5Calpha%20%3D%20%5Cfrac%7BML%5E1T%5E%7B-5%7D%7D%7BML%5E1T%5E%7B-3%7DK%5E%7B-1%7D%7D%20%3D%20T%5E%7B-2%7DK%5E1%0A%5CRightarrow%20%5Cboxed%7B%5BM%5E0L%5E0T%5E%7B-2%7DK%5E1%5D%7D

Final Answer:

%5BM%5E0L%5E0T%5E%7B-2%7DK%5E1%5D

Correct option: (B)

Chapter Questions
1 MCQs