Three long parallel wires are 3.8 cm from one another. (Looking along them, they are at three co

Three long parallel wires are 3.8 cm from one another....

Key Concepts

Magnetic Field Due to a Currents

This concept pertains to the magnetic field created by a long, straight current-carrying conductor, typically given by the formula B = ??I/(2?r). It is fundamental to determining the influence of one wire's current on the surrounding space, which is essential when calculating both forces and fields in multiple wire systems.

Magnetic Force Between Parallel Currents

When two current-carrying wires run parallel to each other, they exert mutual magnetic forces. The force per unit length between them is described by F/L = ??I?I?/(2?r) and depends on the magnitudes of the currents and their separation. This concept is critical for understanding interactions between currents in systems with multiple conductors.

Superposition Principle

In systems with several sources of magnetic fields, the net magnetic field at any given point is the vector sum of the individual fields. This principle is essential when analyzing configurations like an equilateral triangle arrangement of wires, as it allows one to add contributions from each wire to obtain the overall field or force.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of the magnetic field around current-carrying conductors, as well as the direction of the force on a current in a magnetic field. It is indispensable in correctly predicting the orientation of both fields and forces in multi-wire electrical systems.

Transcript

00:03 Okay, so we know the force per unit length can be equal to m .0, i1, i2 over 2pyd.

00:07 So for the wire on the top, this is the 3x diagram for the wire on the top.

00:13 As you can tell, the x components of the magnetic force from other two wires cancel out due to symmetry.

00:21 The angle here is 60 degree, if we divide by 2 is 30 degrees.

00:25 The reason why the angle here is 60 degree here, because 3 wire here, 4 in an equilateral triangle.

00:33 So now, we know the net force on wire m is equal to fnm plus f pm.

00:40 If we expand the equation, we have such equation here.

00:44 And if we do some arrangement here, we have the force per unit length, which is f over lm here, is equal to muil i .n over 2 .0 .m.

00:53 D.

00:53 10x0 .0 .0 is permeability constant, which is equal to 4 pi 10 to the power of 967, tesla, 10 meter per ampere.

01:02 I .n is the current from wire n, i .m is the current from wire m, and ip is the current from yap.

01:11 And we know all of n are equal to a ampere.

01:14 And d here is the separation between each wire, which is 3 .8 centimeter.

01:18 If we convert a meter is equal to 3 .8, that's 10 to the power of negative 2 meter.

01:25 So now if we plug in the equation to determine the force per unit length, we have the value for the force per unit length for wire.

01:34 Wire n is equal to 5 .8 times 10 to the power of negative 4 newton per meter and a direction is point upward.

01:41 So now let's take to get a wire at the left corner which is wire n here.

01:46 And this is the free by diagram for the wire n.

01:51 So the magnitude of the force per unit length for the wire n should be expressed as absolute value f over ln.

02:00 And this can be expressed as such equation here.

02:03 As you can tell, i indicates the x component and y components or such as you can be expressed.

02:07 Force per unit length.

02:09 If we plug in the values, eventually you have the force per unit length for y n is equal to 3 .4 times 10 to the power of a negative 4 newton per meter.

02:19 So now it's determining the direction, which is angle theta here, and this is equal to tension negative 1, f y ln over f x ln.

02:28 So the x components of the force per unit length can be equal to such equation here.

02:35 If we plug in the values, we have the x components of the magnetic force per unit length on wire n is equal to 1 .7 times 10 to the part of negative 4 newton per meter.

02:46 And the y component can be expressed with such equation here.

02:50 If we plug in values, eventually have the y components of force per unit length is equal to negative 2 .9 times 10 to the part of a negative 4 newton per meter.

03:00 So now we can determine direction.

03:02 As you can tell, the y component is having a negative value, but the x components is having a positive value.

03:10 That means the direction of such force per unit length lies along the force quadrants of the xy coordinates.

03:18 So now we have the angle theta here is equal to negative 60 degree based on the calculation.

03:25 So now we can determine the actual angle of the direction for such a force per unit length, which is equal to 360 degree minus 60 degree.

03:35 And this is equal to 300 degree, which means that the direction of such force per unit length is 300 degree above the positive x -axis.

03:49 Another word is 60 degree below the x -axis.

03:55 And that's why we have the angle theta at first is equal to a negative 60 degree.

04:01 So now let's take a look at the different case.

04:07 So for the wire at the right corner, this is the free bi diagram for the wire at the right corner which is wire p here...

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