A non-uniform electric field in x-direction is increasing uniformly from 2N/C at x = 1 m to 8 N/C at x = 7 m . The center of cube is at x = 2 m . If the charge enclosed inside a small cube of side len

Question

A non-uniform electric field in x-direction is increasing uniformly from 2N/C at x = 1 m to 8 N/C at x = 7 m . The center of cube is at x = 2 m . If the charge enclosed inside a small cube of side length 10 cm is 8.85 × 1 0 − 5 n C , the value of n will be.

TestHub · Accepted Answer

The field varies linearly from 2 N/C at x = 1 m to 8 N/C at x = 7 m, so $E (x) = x + 1 N/C$

Cube of side a = 0.1 m is centred at x = 2 m, so its faces are at x = 1.95 m and x = 2.05 m.

Since E is only in the x-direction, net flux comes from these two faces: $ϕ = [E (2.05) - E (1.95)] a^{2} = (3.05 - 2.95) \times (0.1)^{2} = 0.1 \times 0.01 = 1 0^{- 3} N m^{2} /C$

By Gauss law: $q = ε_{0} ϕ = 8.85 \times 1 0^{- 12} \times 1 0^{- 3} = 8.85 \times 1 0^{- 15} C$

Comparing with $8.85 \times 1 0^{- 5 n} C$ : 5n = 15, so n = 3. Answer: 3

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