Question

A 6.0-cm-diameter coil has 50 loops and a resistance of 0.40 12. The central axis of the coil is parallel to the +z-axis, which is perpendicular to, and points out of this page. The coil is situated in a whose 2-component varies with time as B.(t) = (1.0"") (4 – 5 s)? – 2.0 mT. (a) Find an expression for the current, i(t), induced in the coil. (b) Determine the magnitude and direction of the current induced in the coild at t = 2.0 seconds. (c) At which time(s), if any, would the current change direction from that found in part (b)? If the current changes direction, would it continue in this new direction forever, switch back to the direction found in (b), or alternate between the two? Explain your answers, (d) Would any of the results change if the magnetic field had a non-zero component perpendiuclar to the z-axis? Explain your reasoning,

Answer 1

Solution

Induced emf occurs when there is a change in flux per unit time. This emf generates current in the coil

$e=-\frac{\mathrm{d}nBA }{\mathrm{d} t}$

here n is the number of turns

B is magnetic field

A is the area of cross section through which the field is flowing

we will put its respective values at the end

$e=-nA\frac{\mathrm{d}B }{\mathrm{d} t}$

since only the magnetic field is changing

$\Rightarrow e=-nA\frac{\mathrm{d}((t-5)^{2}-2)}{\mathrm{d} t}$

$\Rightarrow e=-nA*2(t-5)$

$\Rightarrow e=-2nA(t-5)$

we know e =iR

R is the resistance

so, current as a function of time is given by

$\Rightarrow i=-\frac{2nA}{R}(t-5)$

putting the values

we get

$\Rightarrow i=-0.706(t-5)\: mA$

at t=2s

i = 2.12 mA clockwise direction

c)

Yes current will change direction when the magnetic field change direction

From the equation we need to find at which time the magnetic field changes towards negative z axis

Since the equation is a parabola the time between the two points when magnetic field becomes zero is negative

B=0

$\Rightarrow (t-5)^{2}-2 =0$

$\Rightarrow t =5\pm \sqrt{2}\: s$

Therefore between this time the current will be anticlockwise and after $t =5+ \sqrt{2}\: s$ it will restore to previous direction

d)

No, it will not make a difference as current or emf induced depends on the change in flux and not flux itself. In order to make a difference it will have to be changing through the area of the loop.

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