First slide

Gauss's law and its applications

Question

It has been experimentally observed that the electric field in a large region of earth's atmosphere is directed vertically down wards. At an altitiude of 300m, the electric field is $60 V / m$ . At an altitude of 200m, the field is $100 V / m$ . Calculate the net amount of charge contained in the cube of 100m edge, located between 200m and 300m altitude

NA

Solution

$φ_{E} = \int_{b o t t o m} \bar{E_{1}} \bar{d s} + \int_{t o p} \bar{E_{2}} \bar{d s} + 4 \int_{v e r t i c a l} \bar{E} \bar{d s} φ_{E} = \int_{b o t t o m} E_{1} d s \cos o^{0} + \int_{t o p} E_{2} d s \cos 180^{0} + 4 \int_{v e r t i c a l} E d s \cos 90^{0} φ_{E} = (E_{1} - E_{2}) s φ_{E} = \frac{q}{\in_{0}} \Rightarrow q = φ_{E} \in_{0} q = (E_{1} - E_{2}) s \in_{0} = (100 - 60) {(100)}^{2} (8.85 \times 10^{- 12}) = 3.54 \times 10^{- 6} C$

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