Ampere's Law and Walter Lewin

[flyingpig]

Homework Statement

http://img18.imageshack.us/img18/8196/ampere.th.png

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The Attempt at a Solution

I've watched Walter Lewin's vid http://www.youtube.com/watch?v=sxCZnb-EMtk&feature=relmfu like five times and he seemed pretty angry with the way books explain this...

Anyways

First and foremost

$$\oint \vec{B} \cdot \vec{ds} = \mu_0 I$$

(a) my Amperian loop encloses no current, so it is 0

(b)http://img716.imageshack.us/img716/339/ampb.th.png

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By the right hand rule, my thumb points into the page and my B-field vector points as in the picture.

So I get

$$\oint \vec{B} \cdot \vec{ds} = \mu_0 I_1 N_1$$

$$B(2\pi b) = \mu_0 I_1 N_1$$

$$B = \frac{ \mu_0 I_1 N_1}{2\pi b}$$

Now for c, I got to enlarge my Amperian loop

http://img17.imageshack.us/img17/6403/ampc.th.png

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Since the currents are in opposite direction, I must take their absolute value difference. So I have something like this

$$\oint \vec{B} \cdot \vec{ds} = \mu_0 NI$$

$$B(2\pi c) = \mu_0 \left |N_1I_1 - N_2I_2 \right|$$

$$B = \frac{\mu_0 \left |N_1I_1 - N_2I_2 \right|}{2\pi c}$$

I am actually pretty confident about this, but I just started this today, so I need a thumbs up from an expert

 

[ideasrule]

Your answers are correct. Note, though, that they are not exact, because the configuration is not exactly symmetrical. For instance, the field very near a wire will be dominated by that wire, while the field far from any wire will be close to the answers you got.

 

[flyingpig]

What do you mean not symmetrical? I used a circle as my closed path...circles are very very symmetric!

 

[ideasrule]

Yes, but the actual wires do not form a perfect circle, because there are gaps in between. However, I'm very sure that they don't matter for the purposes of this question, and that your answers are the intended ones.

 

[rwisz]

I would like to clarify and expand on the concepts of Ampere's Law and Walter Lewin's explanation of it.

Ampere's Law is a fundamental principle in electromagnetism that relates the magnetic field around a closed loop (represented by the integral of B·ds) to the current enclosed by that loop (represented by μ0I). This law is crucial in understanding the behavior of magnetic fields and is often used in the design and analysis of electrical systems.

In the video by Walter Lewin, he highlights the importance of understanding the direction of the magnetic field and current when applying Ampere's Law. This is crucial because the direction of the magnetic field and current can affect the sign of the enclosed current, which in turn affects the magnitude of the magnetic field.

In part (a) of the problem, the Amperian loop encloses no current, and therefore the integral of B·ds is equal to zero. This is because Ampere's Law states that the magnetic field around a closed loop is directly proportional to the current enclosed by that loop. In this case, since there is no enclosed current, there is no magnetic field.

In part (b) of the problem, the right-hand rule is used to determine the direction of the magnetic field. This is a common method used in electromagnetism to determine the direction of the magnetic field based on the direction of the current. The equation used in this part is a simplified version of Ampere's Law, where the number of turns (N) and the current (I) are multiplied together to represent the total enclosed current.

In part (c), the Amperian loop is enlarged to include two currents flowing in opposite directions. In this case, the enclosed current is calculated by taking the absolute value of the difference between the two currents. This is because the direction of the magnetic field is affected by the direction of the current, and the direction of the current will be different for each current flowing in opposite directions.

In conclusion, Ampere's Law is a fundamental principle in electromagnetism that relates the magnetic field around a closed loop to the current enclosed by that loop. It is important to understand the direction of the magnetic field and current when applying this law, as highlighted by Walter Lewin in his explanation. By carefully considering the direction of the magnetic field and current, we can accurately calculate the magnitude of the magnetic field using Ampere's Law.