A length –scale (l) depends on the permittivity $$($$ε$$)$$ of a dielectric material ,Boltzmann constant (kB) , the absolute temperature (T) , the number per unit volume (n) of certain charged particles , and the charge (q) carried by each of the particles. Which of the following $$expression(s)$$ for l is (are) dimensionally correct?

Question

A  length –scale (l) depends on the  permittivity $$($$ε$$)$$      of a  dielectric material ,Boltzmann constant (kB)     , the absolute temperature (T) , the number per unit  volume (n) of certain charged particles , and the charge (q) carried by each  of the particles. Which of the following $$expression(s)$$ for l is (are) dimensionally correct?

Infinity Learn · Accepted Answer

Permittivity ε$$=M$$−$$1L$$−$$3T4I2$$ Boltzmen constant $$KB=M1L2T$$−$$2K$$−$$1Temperature$$ T $$=$$ KNo.of particles per unit $$volumen=L$$−$$3Charge$$ $$q=I1T1For$$ A: $$nq2$$ε$$KBT=L$$−$$3I2T2M$$−$$1L$$−$$3T4I2M1L2T$$−$$2K$$−$$1k=1L=L$$−$$1For$$ B: ε$$KBTnq2=L$$            [ from option A]For C: $$q2$$ε$$n2/3KBT=I2T2M$$−$$1L$$−$$3T4I2L$$−$$2ML2T$$−$$2K$$−$$1K=L3/2For$$ D: $$q2$$ε$$n1/3KBT=I2T2M$$−$$1L$$−$$3T4I2L$$−$$1ML2T$$−$$2K$$−$$1k=L$$

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