Determine the magnetic field at the point P due to a very ong wire with a square bend as shown i

Determine the magnetic field at the point P due to a very...

Key Concepts

Symmetry in Magnetic Field Calculations

Exploiting symmetry in a system significantly simplifies the evaluation of magnetic fields. In arrangements where the geometry or current distribution has symmetrical features, certain field components can cancel or combine in predictable ways. Recognizing such symmetry helps in reducing the complexity of the integrals involved and in understanding the overall field configuration.

Superposition Principle

The superposition principle in electromagnetism states that the total magnetic field created by multiple current-carrying segments is the vector sum of the magnetic fields produced by each individual segment. This concept allows complex geometries, such as wires with bends or multiple segments, to be analyzed by calculating the field due to each segment separately and then combining them.

Biot–Savart Law

The Biot–Savart Law is a fundamental principle in magnetostatics that relates a small segment of current to the magnetic field it produces. It states that each element of current creates a magnetic field, the magnitude and direction of which depend on the distance from the element as well as the orientation of the element relative to the point at which the field is being calculated. This law is used to derive the magnetic field for wires of arbitrary shape through integration over the current distribution.

Magnetic Field Due to a Finite Current Segment

When calculating the magnetic field from a segment of current-carrying wire that is not infinitely long, the field can be determined by integrating the contributions of each current element along the segment. This often involves expressing the field in terms of the angles that the segment subtends at the point of interest, leading to an expression that depends on the geometry of the configuration.

Transcript

00:01 For this question, we're asked to determine the magnetic field at point p in the figure 60, where point p is halfway between the two corners.

00:09 And we can use this to determine the results of the problem for 40 and 41 .1.

00:14 So if you don't know those results are unfamiliar, you can work those out or go back and view the videos on those because i'm going to be utilizing them here.

00:21 So the total field b is going to be equal to b1 plus b2 plus b3, the contribution of these three different points.

00:27 So we simply have to use the results from these previous questions to find.

00:31 These answers.

00:31 So b1 here is going to be mu not magnetic permeability of free space times the current i divided by 4 pi times the area a, or excuse me, the distance a from the figure over 2.

00:44 This gets multiplied by the sign of the angle theta plus 2 divided by over 2 times root 5.

01:03 We can simplify this a little bit, and what we find is this is equal to mu not times i divided by 2 pi a multiplied by 2 divided by the square root of 5 b2 we find is equal to mu not times i divided by 4 pi this time is just times a not a over 2 and this is multiplied by the first value is a over 2 divided by a over 2 times root 5 and then this gets added to the second value which is the exact same.

02:01 It's a over 2 divided by a over 2 times root 5.

02:09 Simplifying this a little bit, we find that this comes out to equal.

02:13 We have mu not i over 2 pi a, and then everything in the parenthesis simplifies down to 1 over root 5.

02:31 Almost done...

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