Final answer:
The induced current at time t = 5s is 1.6 A, and by Lenz's Law, its direction is such that it creates a magnetic field out of the page, opposing the decrease in the external magnetic field.
Step-by-step explanation:
To find the magnitude and direction of the induced current in the wire loop with a time-dependent magnetic field, we apply Faraday's Law of Electromagnetic Induction. Faraday's Law states that the induced electromotive force (emf) in a closed loop equals the negative change in magnetic flux through the loop per unit time, ℑ = -dΦ/dt.
First, we calculate the rate of change of the magnetic field B(t) given by B(t) = 250 - 4t2, at the time t = 5s:
dB/dt = d/dt(250 - 4t2) = -8t
At t = 5s, dB/dt = -8(5) = -40 T/s.
Now we'll use the induced emf ℑ formula:
ℑ = -A * dB/dt
Where:
- A is the area of the wire loop (4.0 m2).
- dB/dt is the rate of change of the magnetic field strength.
ℑ = -4.0 * (-40 T/s) = 160 V
The induced current I can be found using Ohm's Law, I = ℑ/R.
I = 160 V / 100 Ω = 1.6 A
The direction of the induced current is determined by Lenz's Law, which states that the direction of the induced current will be such that it opposes the change in magnetic flux. Since the magnetic field is decreasing, the induced current will be in a direction that would create a magnetic field out of the page, which supports the original magnetic field that is weakening.