Tentukan medan magnet (dalam , , dan ) pada pusat koordinat yang disebabkan
karena kawat berarus seperti pada. Gambar 7. Solusi: Untuk mengerjakan soal ...
Soal: Tentukan medan magnet (dalam , , dan ) pada pusat koordinat yang disebabkan karena kawat berarus seperti pada Gambar 7.
Solusi: Untuk mengerjakan soal ini, cari dulu bentuk umum medan magnet yang ditimbulkan oleh kawat berarus. Cek dokumen perkuliahan 11 - Medan Magnet oleh Arus Listrik.pdf halaman 10
catatan penting : Untuk persamaan Biot-Savart berikut :
̂
1. Arah vektor ̂ berpangkal pada elemen kawat (sumber) dan menuju titik pengamatan . 2. Sudut pada persamaan tersebut adalah sudut yang dibentuk antara dan ̂ .
S ECTI O N 3 0.1 • The Biot–Savart Law
Example 30.1
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Interactive
Magnetic Field Surrounding a Thin, Straight Conductor
Consider a thin, straight wire carrying a constant current I and placed along the x axis as shown in Figure 30.3. Determine the magnitude and direction of the magnetic field at point P due to this current.
product is simply the magnitude of ds, which is the length dx. Substitution into Equation 30.1 gives
Solution From the Biot–Savart law, we expect that the magnitude of the field is proportional to the current in the wire and decreases as the distance a from the wire to point P increases. We start by considering a length element d s located a distance r from P. The direction of the magnetic field at point P due to the current in this element is out of the page because ds ⫻ ˆr is out of the page. In fact, because all of the current elements I ds lie in the plane of the page, they all produce a magnetic field directed out of the page at point P. Thus, we have the direction of the magnetic field at point P, and we need only find the magnitude. Taking the origin at O and ˆ being a letting point P be along the positive y axis, with k unit vector pointing out of the page, we see that
Because all current elements produce a magnetic field ˆ direction, let us restrict our attention to the in the k magnitude of the field due to one current element, which is
d s ⴛ ˆr d s ⴛ ˆr kˆ (dx sin ) kˆ where d s ⴛ ˆr represents the magnitude of ds ⴛ ˆ r. Because ˆ r is a unit vector, the magnitude of the cross
ˆ 0I dx sin k ˆ d B (dB) k 4 r2
(1)
dB
To integrate this expression, we must relate the variables , x, and r. One approach is to express x and r in terms of . From the geometry in Figure 30.3a, we have (2)
r
d s = dx
x a cot Taking the derivative of this expression gives dx a csc2 d
Substitution of Equations (2) and (3) into Equation (1) gives
P
(4) r rˆ
a
θ x ds
a a csc sin
Because tan a/( x) from the right triangle in Figure 30.3a (the negative sign is necessary because d s is located at a negative value of x), we have
(3) y
0 I dx sin 4 r2
B
I
0I a csc 2 sin d 0I sin d 4 a 2 csc 2 4a
an expression in which the only variable is . We now obtain the magnitude of the magnetic field at point P by integrating Equation (4) over all elements, where the subtending angles range from 1 to 2 as defined in Figure 30.3b:
O x
dB
0I 4a
2
sin d
1
0I (cos 1 cos 2) 4a
(30.4)
(a)
P
θ2
θ1
We can use this result to find the magnetic field of any straight current-carrying wire if we know the geometry and hence the angles 1 and 2. Consider the special case of an infinitely long, straight wire. If we let the wire in Figure 30.3b become infinitely long, we see that 1 0 and 2 for length elements ranging between positions x and x . Because (cos 1 cos 2) (cos 0 cos ) 2, Equation 30.4 becomes B
(b)
Figure 30.3 (Example 30.1) (a) A thin, straight wire carrying a current I. The magnetic field at point P due to the current in each element ds of the wire is out of the page, so the net field at point P is also out of the page. (b) The angles 1 and 2 used for determining the net field. When the wire is infinitely long, 1 0 and 2 180°.
0I 2a
(30.5)
Equations 30.4 and 30.5 both show that the magnitude of the magnetic field is proportional to the current and decreases with increasing distance from the wire, as we expected. Notice that Equation 30.5 has the same mathematical form as the expression for the magnitude of the electric field due to a long charged wire (see Eq. 24.7).
At the Interactive Worked Example link at http://www.pse6.com, you can explore the field for different lengths of wire.
