A loop of wire in the shape of a rectangle of width w and...

A loop of wire in the shape of a rectangle of width $w$ and length $L$ and a long, straight wire carrying a current $I$ lie on a tabletop as shown in Figure P31.11. (a) Determine the magnetic flux through the loop due to the current $I$ (b) Suppose the current is changing with time according to $I=a+b t,$ where $a$ and $b$ are constants. Determine the emf that is induced in the loop if $b=10.0 \mathrm{A} / \mathrm{s}, h=1.00 \mathrm{cm},$ $w=10.0 \mathrm{cm},$ and $L=1.00 \mathrm{m} .$ (c) What is the direction of the induced current in the rectangle?

Key Concepts

Lenz’s Law

Lenz's Law is a principle that dictates the direction of the induced current resulting from a changing magnetic flux. It states that the induced current will flow in such a direction that its own magnetic field opposes the change in the original magnetic flux. This self-correcting behavior ensures the conservation of energy in electromagnetic systems.

Magnetic Flux

Magnetic flux is defined as the product of the magnetic field and the area through which the field lines pass, taking into account the angle between the field and the normal to the area. It is a measure of the 'amount' of magnetic field that penetrates a given area and is calculated by integrating the magnetic field over the surface area. This concept is crucial for understanding how changing magnetic fields can induce electric currents in loops.

Faraday’s Law of Induction

Faraday’s Law of Induction states that a changing magnetic flux through a circuit induces an electromotive force (emf) in the circuit proportional to the rate at which the flux changes. This law forms the basis for many electrical devices and is mathematically expressed as emf = -d?/dt, where the negative sign indicates the direction of the induced emf given by Lenz's Law.

Magnetic Field of a Long Straight Conductor

The magnetic field around a long straight current-carrying conductor is described by Ampère's law and is given by an expression that shows the field decreases with distance from the wire. This inverse relationship with distance is important for calculating the magnetic flux through structures placed near a current-carrying wire.

Transcript

00:01 In the given problem, we have been given a straight wire carrying a current i in it from left to right and a rectangular lou having bit w and lamp l.

00:29 The upper arm of the loop is at a distance edge from the wire.

00:36 In the first part of the problem, we have to find the total magnetic flux passing through this loop the cannula loop as the magnetic field.

00:45 Due to this current carrying conductor, we'll keep changing.

00:50 The magnetic field depends on the distance, and the distance of various segments of this loop from the wire are changing, so the flux will keep changing.

01:01 So to find their total flux, we will use calculus methods, means differentiation and integration.

01:08 So, first of all, we consider a small strip of this rectangular loop whose land is same as l.

01:20 But licked is bs, and it is at a distance of x from the upper side of the loop, so its distance from the wire will become edge plus x.

01:46 An area small area of this strip will become with b x and limp.

01:57 Any means l d x.

02:01 So we know that it stands so at this distance from the wire.

02:08 The magnetic field will be given us expression for magnetic field traditions are from a straight current carrying conductor is given as new not upon four pi to i buy art so here it will become you're not by two by i buy ahh plus x.

02:37 So the small flux we will consider this magnetic field as constant, which will be passing through this segment as the distance dx is negligible small.

02:49 So there will be no change in the magnetic field passing through this small strip.

02:54 Hence, this small flux will become b into dear means new not by two pi into i by h plus x and for area and dx.

03:22 So if you want to find that total flux, we will integrate the expression so total magnetic flux will be given ass integration of this.

03:35 Yeah, we're not upon two pi i l d.

03:40 It's edge plus x here.

03:45 The constants will come out so it will become you not i l by two pi and very many will be won by h plus x dx.

04:00 So, as we have to integrate this ex and the x is wearing from zero to the maximum w so the limits of this integration will be from zero to w.

04:13 Now the integration of one upon a regular sex with respect to the years comes out to be log b c e edge plus x limits zero to w.

04:35 And then i put these limits.

04:37 I'll get long based e h plus w minus log.

04:52 Aah! means when i put the upper limit w the ex will be in nsw and in the lower limit, zero x will be reading a zero so it will win, actually lot busy.

05:08 So finally the expression for the flux becomes new, not i, and by two pi love this evening's natural algorithm h last w by edge...

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