电偶极子论文

Electric Dipole’s R﹑d Orders Of Magnitude Conditions

By KanSen

In the calculation of the electric intensity E or potential V of a electric dipole, If meet R > > d, We often take the approximate calculation to simplify calculations, But, only when what order of magnitude condition of R and d is meeted, can we use approximate calculation ,whose error is withen our acceptable range. Then we will analysise this problem;

In order to study the effect of the relationship of R and D orders of magnitude on the error of the approximate calculation, We first use the accurate calculation formula to calculate the potential V of a electric dipole on a certain point. Then we use the approximate calculation formula to calculate the potential on that point.Next we can observe the difference between the theoretical result and approximate result.

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Electric dipole model

Accurate calculation formula: V=)11(4-+-R R q

o πε ,

R + =【(Rsin θ)2 +(Rcos θ-d/2)2】1/2

R -=【(Rsin θ)2 +(Rcos θ+d/2)2】1/2 V=o q

πε4{【(Rsin θ)2 +(Rcos θ-d/2)2】-1/2 -【(Rsin θ)2 +(Rcos θ+d/2)2】-1/2}

If R>>d,we have the following trasformation:

Approximate calculation:

1/R +≈(R-dcos θ/2)-1≈1/R(1+ dcos θ/(2R))

1/R -≈(R+dcos θ/2)-1≈1/R(1- dcos θ/(2R))

V ’=24cos R qd o πεθ

Now, let us assume that q=1×10-6 C, d=1×10-5 m, θ=π/4 rad, when R Takes different values , we can get the corresponding value of V and V ’, As listed in the following table:

From the table we can probably come to the conclusion that: when the difference of orders of magnitude of R and d was greater than 3, We can use the approximation but will not Have much error.

The equipotential surface in the electric field of an electric dipole:

(Approximatedly)

Reference:

Field and Wave Electromagnetics [David K.Cheng] V>0

V<0