Flux due to wire ...What is the magnetic flux through the loop at time t= 2.00 seconds? (Define positive flux into the page.) please helpp?
A rectangular loop with sides of length a= 1.30 cm and b= 3.70 cm is placed near a wire that carries a current that varies as a function of time: it=3.67 + 1.12t2 where the current is in Amperes and the time is in seconds. The distance from the straight wire to the closest side of the loop is d= 0.420 cm.
a)What is the magnetic flux through the loop at time t= 2.00 seconds? (Define positive flux into the page.)b)What is the induced e.m.f. in the loop at time t= 2.00 seconds? (Note that positive emf is clockwise.)
- ?Lv 75 months ago
Thank you for Best Answer to the other version of this question! I can't edit it now, so here's my worked solution.
A. You can’t just multiply the area by B to get the flux because B is not constant across the area. And you can’t use a simple average because the field does not vary linearly with distance from wire. B is stronger nearer the wire according to B = μ₀i/(2πr). Integration is needed.
We break the area into strips parallel to the wire. Each strip is length 0.037m and width dr, so area of each strip is dA=0.037dr. Flux through the strip is BdA:
dΦ = BdA
= (μ₀i/(2πr)) 0.037dr
= (4π*10^-7i/(2πr)) * 0.037dr
= 7.4*10^-9 i dr/r
To get total flux through loop, we integrate dΦ from r = 0.0042m to r = 0.00420+0.0130 = 0.0172m
Φ = ∫dΦ
= 7.4*10^-9 i ∫r/dr (for r=0.0042m to 0.0172m)
= 7.4*10^-9 i [ln(r)]
= 7.4*10^-9 i [ln(0.0172) – ln(0.0042)]
= 7.4*10^-9 i [1.4098]
= 1.0432*10^-8 i (equation 1)
i(t)=3.67 + 1.12t²
When t=2.00s, i = 3.67 + 1.12(2.00)² = 8.15A
From equation 1:
Φ = 1.0432*10^-8 * 8.15 = 8.502*10^-8 Wb
Applying the right hand grip rule to the wire, the direction of B (hence direction of Φ) is into the page. So the flux is positive and to 3 sig. figs. is 8.50*10^-8 Wb.
B. The magnitude of the induced emf is |emf| = |dΦ/dt|.
From equation 1:
dΦ/dt = 1.0432*10^-8 di/dt
di/dt = 2.24t. When t=2.00s, di/dt = 2.24*2.00 = 4.48 A/s
|emf| = 1.0432 * 10^-8 * 4.48 = 4.67*10^-8 V
The flux is into the page and increasing (as i increases with time). So the induced current produces an opposing field directed out of the page. Applying right hand rule, current in loop is anticlockwise. So emf is negative, i.e. -4.67*10^-8 V
Arithmetic no guaranteed though.
- FiremanLv 75 months ago
Kindly attach the diagram