(a) A conducting loop in the shape of a square of edge length ℓ=0.400 m carries a current I=10.0

(a) A conducting loop in the shape of a square of edge...

(a) A conducting loop in the shape of a square of edge length $\ell=0.400 \mathrm{m}$ carries a current $I=10.0 \mathrm{A}$ as shown in Figure $\mathrm{P} 30.5 .$ Calculate the magnitude and direction of the magnetic field at the center of the square. (b) What If? If this conductor is reshaped to form a circular loop and carries the same current, what is the value of the magnetic field at the center?

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle in electromagnetism that provides a method to calculate the magnetic field produced at a point in space by a differential segment of a steady current. It involves integrating these contributions over the entire current-carrying conductor and is key to solving for the magnetic field in various geometries.

Magnetic Field of a Current Loop

This concept involves determining the magnetic field at specific points, such as the center of a loop, by summing the contributions of all current elements forming the loop. Analytic formulas exist for standard shapes like circular loops, and similar techniques using symmetry and integration can be applied to other shapes such as squares.

Symmetry in Magnetic Field Calculations

Utilizing symmetry in problems involving current loops simplifies the calculation of the magnetic field. In symmetric configurations, contributions from different current segments can combine in a consistent direction, allowing for the derivation of compact expressions for the net field at points of symmetry, such as the center of the loop.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around a current-carrying conductor. By orienting the thumb in the direction of the current flow, the curl of the fingers indicates the direction in which the magnetic field lines circulate, which is critical for understanding the vector nature of the magnetic field.

Transcript

00:01 In this problem, we have a conducting loop in the shape of a square with given edge length, and we want to calculate the magnitude and direction of the magnetic field at the center of that square.

00:10 We then want to find what would happen if this conductor was reshaped to be a circular loop and carry the same current.

00:16 So in the first case, we know that we can rewrite our typical equation for b if we are working at a point that is a certain distance from our current carrying wire.

00:25 In this case, our b is not equal to mu not i over 2 pi r, but equal to mu not i over a 4.

00:31 4 pi r times sine of theta i plus sine of theta 2.

00:34 And our thetas are labeled in this diagram over here.

00:38 Now in this problem, we are given that the length of 100 of the square is 0 .4 meters and the current is 10 a peers.

00:46 So, and we're also given the fact that since this is a square, all of our thetas must be equal to 45 degrees.

00:53 Furthermore, since we're finding the curve, this is furthermore, since we're finding the magnitude direction of the field at a, at a, at the other corner of the square, and we know that a diagonal of a square is its length divided by 2, we also have a, our radius.

01:09 Now, we can plug in all the quantities that we need to solve our problem, which we find then is equal to the square root of 2 times mu not times i divided by 2 pi l, or l is the length of one side of our square.

01:22 We also find, using the right hand rule, which you can visualize on your own, that the field must be pointing into the page.

01:29 Now, a square has four sides, obviously, so our magnetic field isn't just coming from one of them...

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