The magnetic flux through a stationary loop with a resistance R varies during interval of time T as φ=a t(T- t). If the heat generated during this time, neglecting the inductance of the loop, is (a2 T3/p R), then find p.

Question

The magnetic flux through a stationary loop with a resistance R varies during interval of time T as φ=a t(T- t). If the heat generated during this time, neglecting the inductance of the loop, is (a2 T3/p R), then find p. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

φ=a t(T-t) ; |ε|=(d φ/a t)=a T-2 a t d H=ε I d t=((a T-2 a t)(a T-2 a t)/R) d t ∫ limits0H d H=∫ limits0T ((a T-2 a t)2/R) d t H=(a2 T3/3 R)

Q. The magnetic flux through a stationary loop with a resistance $R$ varies during interval of time $T$ as $ϕ = a t (T -$ $t)$ . If the heat generated during this time, neglecting the inductance of the loop, is $\frac{a ^{2} T ^{3}}{pR}$ , then find $p$ .

$ϕ = a t (T - t); ∣ ε ∣ = \frac{d ϕ}{a t} = a T - 2 a t$
$d H = ε I d t = \frac{( a T - 2 a t ) ( a T - 2 a t )}{R} d t$
$0 \int H d H = 0 \int T \frac{( a T - 2 a t ) ^{2}}{R} d t$
$H = \frac{a ^{2} T ^{3}}{3 R}$

Q. The magnetic flux through a stationary loop with a resistance R varies during interval of time T as ϕ=at(T− t). If the heat generated during this time, neglecting the inductance of the loop, is pRa2T3​, then find p.

ϕ=at(T−t);∣ε∣=atdϕ​=aT−2at dH=εIdt=R(aT−2at)(aT−2at)​dt 0∫H​dH=0∫T​R(aT−2at)2​dt H=3Ra2T3​

Q. The magnetic flux through a stationary loop with a resistance $R$ varies during interval of time $T$ as $ϕ = a t (T -$ $t)$ . If the heat generated during this time, neglecting the inductance of the loop, is $\frac{a ^{2} T ^{3}}{pR}$ , then find $p$ .

$ϕ = a t (T - t); ∣ ε ∣ = \frac{d ϕ}{a t} = a T - 2 a t$
$d H = ε I d t = \frac{( a T - 2 a t ) ( a T - 2 a t )}{R} d t$
$0 \int H d H = 0 \int T \frac{( a T - 2 a t ) ^{2}}{R} d t$
$H = \frac{a ^{2} T ^{3}}{3 R}$