analyze the dipole for dipole moment in azdirection.

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Course: Electromagnetic Field Theory

 

An electric dipole is located at (1, 1, 0). If the voltage at

(5, 6, 0) is 9V analyze the dipole for dipole moment in azdirection.

Expert Solution
Step 1

Electric Dipole

If two charges of equal magnitude and opposite signs are placed at a very small distance from one another, then the system is known as an electric dipole. The strength of an electric dipole is measured by its dipole moment vector. If the magnitude of the charge is q and the separation between the charges is 2a then the dipole moment is given by

p=q.2ap^

The direction of the dipole moment is from the negative to the positive charge.

Let us consider two charges q and -q be placed at a distance of 2a. Let r be the distance between the center of the dipole and the point of interest P. Let r1 and r2 be the distance between the point P and +q and -q respectively. 

Advanced Physics homework question answer, step 1, image 1

Potential due to charge -q at P=14πε0-qr2

Potential due to charge +q at P=14πε0qr1

The total potential at P is

V=14πε0qr1-14πε0qr2=14πε01r1-1r2

Applying cosine law of vector we have

r12=r2+a2-2racosθr1=r2+a2-2racosθ12r1=r1+a2r2-2arcosθ12

Now for a short dipole, we have

r1=r1-2arcosθ121r1=1r1-2arcosθ-121r1=1r1+arcosθ

Similarly, we can see that

1r2=1r1-arcosθ

Therefore the potential 

V=q4πε01+arcosθ-1+arcosθ=q×2acosθ4πε0r2=14πε0pcosθr2=14πε0p.r^r2

 

 

Step 2

In the question given, the dipole is located at the position i^+j^ and the position of the observation point is 5i^+6j^. Therefore

r=5i^+6j^-i^+j^=4i^+5j^

Therefore

r2=42+52=16+25=41

and 

r^=rr=441i^+541j^

Let the dipole moment is given by

p=pxi^+pyj^+pzk^

Therefore we have

9=14πε04px41+5py4141

Thus we wee that there is no component of the dipole moment in the z-direction, that is, pz=0.

Considering the dipole is kept on the x axis. Therefore

r.i^=rcosθcosθ=441

Therefore the magnitude of the dipole moment

9=14πε0p44141p=9×4π×8.85×10-12×41×414Cm=6.56×10-8 Cm

 

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