Observe that the given vector field is a gradient field:
Let
, so that



Integrating the first equation with respect to
, we get

Differentiating this with respect to
gives

Now differentiating
with respect to
gives

Putting everything together, we find a scalar potential function whose gradient is
,

It follows that the curl of
is 0 (i.e. the zero vector).
Answer:
the time taken t is 9.25 minutes
Explanation:
Given the data in the question;
The initial charge on the supercapacitor = 2.1 × 10³ mV = 2.1 V
now, every minute, the charge lost is 9.9 %
so we need to find the time for which the charge drops below 800 mV or 0.8 V
to get the time, we can use the formula for compound interest in basic mathematics;
A = P × ( (1 - r/100 )ⁿ
where A IS 0.8, P is 2.1, r is 9.9
so we substitute
0.8 = 2.1 × ( 1 - 0.099 )ⁿ
0.8/2.1 = 0.901ⁿ
0.901ⁿ = 0.381
n = 9.25 minutes
Therefore, the time taken t is 9.25 minutes
Answer:
(B) The total internal energy of the helium is 4888.6 Joules
(C) The total work done by the helium is 2959.25 Joules
(D) The final volume of the helium is 0.066 cubic meter
Explanation:
(B) ∆U = P(V2 - V1)
From ideal gas equation, PV = nRT
T1 = 21°C = 294K, V1 = 0.033m^3, n = 2moles, V2 = 2× 0.033=0.066m^3
P = nRT ÷ V = (2×8.314×294) ÷ 0.033 = 148140.4 Pascal
∆U = 148140.4(0.066 - 0.033) = 4888.6 Joules
(C) P2 = P1(V1÷V2)^1.4 =148140.4(0.033÷0.066)^1.4= 148140.4×0.379=56134.7 Pascal
Assuming a closed system
(C) Wc = (P1V1 - P2V2) ÷ 0.4 = (148140.4×0.033 - 56134.7×0.066) ÷ 0.4 = (4888.6 - 3704.9) ÷ 0.4 = 1183.7 ÷ 0.4 = 2959.25 Joules
(C) Final volume = 2×initial volume = 2×0.033= 0.066 cubic meter