Question

A stiff wire 50.0 cm long is bent at a right angle in the middle. One section lies along the z axis and the other is along the line y=2x in the xy plane. A current of 20.0 A flows in the wire --- down the z axis and out the line in the xy plane. The wire passes through a uniform magnetic field B = (.318 i)T. Determine the magnitude and the direction of the total force on the wire.

Answer 1

Magnitude of the force is 2.135N and the direction is 41.8° below negative y-axis

Step-by-step explanation:

The stiff wire 50.0cm long bent at a right angle in the middle

One section lies along the z axis and the other is along the line y=2x in the xy plane

`(y)/(x) = 2`

tan θ = 2

Therefore,

slope m = tan θ = y / x

`\theta=\tan^-^1(2)=63.4^0`

Then length of each section is 25.0cm

so, length vector of the wire is

`\hat I= (-25.0)\hat k +(25.0) \cos 63.4^0 \hati +(25.0) \sin63.4^0 \hatJ\\\\\hat I = (11.2) \hat i + (22.4) \hat j - (23.0) \hat k`

And magnetic field is B = (0.318T)i

Therefore,

`\bar F = \hat I (\bar l * \bar B)`

`\bar F = (20.0)[(0.112m)i +(0.224m)j-(0.250m)k * 90.318T)i]`

`= (20.0)(i(0)+j(-0.250)(0.318T)+k[0-(0.224m)(0.318T)]\\\\=(20.0)(-0.250)(0.318)j-(20.0)(0.224)(0.318T)\\\\=-(1.59N)j-(1.425N)k`

Magnitude of the force is

`F = √((-1.59N)^2+(-1.425N)^2\\) \\F = 2.135N`

Direction is

`\alpha = \tan^-^1((-1.425N)/(-1.59N) )\\\\= 41.8^0`

Magnitude of the force is 2.135N and the direction is 41.8° below negative y-axis

Answer 2

F=2.38i - 4.57j

magnitude of F=5.15N

Step-by-step explanation:

The force on the wire is given by

`F=Il \ X B`

in this problem we have to compute the force for both sections if the wire. Hence we have

`F=Il_1\ X B+Il_2\ X B\\\\`

The direction of l1 is k. The direction of l2 is obtained by using the slope of the line y=2x

`tan\theta =2\\\theta=63.43\°\\l_2=(0.5cos(63.43))\hat{i}+(0.5sin(63.43))\hat{j}=0.22\hat{i}+0.89\hat{j}`

By applying the cross product we have

`F=-(20.0A)(0.5m)(0.318T)\hat{j}+(20.0A)(0.445m)(0.318T)\hat{i}-(20.0A)(0.22m)(0.318T)\hat{j}\\\\F=-3.18N\hat{j}+2.83N\hat{i}-1.39N\hat{j}=2.38N\hat{i}-4,57N\hat{j}`

and its magnitude is

`|F|=√((2.38)^2+(4.57)^2)=5.15N`

HOPE THIS HELPS!!