An electric dipole consists of two electric charges of equal...

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are $ q $ and $ -q $ and are located at a distance $ d $ from each other, then the electric field $ E $ at the point $ P $ in the figure is $ E = \frac {q}{D^2} - \frac {q}{(D + d)^2} $ By expanding this expression for $ E $ as a series in powers of $ d/D, $ show that $ E $ is approximately proportional to $ 1/D^3 $ when $ P $ is far away from the dipole. $ \mid \longleftarrow D \longrightarrow \mid \leftarrow d \rightarrow \mid $

An electric dipole consists of two electric charges of equal magnitude and opposite sign. If the charges are $ q $ and $ -q $ and are located at a distance $ d $ from each other, then the electric field $ E $ at the point $ P $ in the figure is
$ E = \frac {q}{D^2} - \frac {q}{(D + d)^2} $
By expanding this expression for $ E $ as a series in powers of $ d/D, $ show that $ E $ is approximately proportional to $ 1/D^3 $ when $ P $ is far away from the dipole.
$ \mid \longleftarrow D \longrightarrow \mid \leftarrow d \rightarrow \mid $

Key Concepts

Far-Field Approximation

The far-field approximation involves analyzing the behavior of a field at distances large compared to the size of the source, in this case, the dipole separation. Under this approximation, higher-order terms in the series expansion become negligible, and the leading term—often responsible for the dominant behavior (here, 1/D^3 decay of the dipole's electric field)—provides the essential physical insight.

Series Expansion

Series expansion, such as a Taylor or binomial expansion, is a mathematical technique used to approximate functions by expressing them as a power series in a small parameter. In many physical contexts, including the analysis of the dipole field at large distances, this method simplifies complex expressions such that terms can be grouped according to their order of magnitude.

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a finite distance. It is a key concept in electromagnetism, providing a simplified model to analyze how separated charges produce an electric field, especially in systems where the positive and negative charges are spatially displaced.

Dipole Moment

The dipole moment is a vector quantity defined as the product of the magnitude of one of the charges and the distance separating the charges. It characterizes the strength and orientation of a dipole, fundamentally influencing the resulting distribution of the electric field in space.

Electric Field

The electric field is a vector field that represents the force per unit charge exerted on a small test charge placed in the vicinity of other charges. In the context of a dipole, the field arises from the combined contributions of both charges and exhibits a distinct spatial dependence that can be derived using principles of vector superposition.

Transcript

00:01 In this problem we're given a scenario, we're given a positive charge and a negative charge where the distance between those two is lowercase d and there's another particle that is located the units away from that dipole on electric field for such a scenario is given as q over d squared minus q over d plus d squared and we're asked to expand this electric field as a power of d over d so lowercase over uppercase d and we want to show that electric field is proportional to 1 over uppercase d cubed.

00:36 All right.

00:37 Now, since we want to show that that as a function of d over d ratio of the distances, let's write this one first as electric field is equal to q over capital d squared minus q over d plus d squared.

00:53 That is equal to q over d squared minus q over d squared 1 plus d over d squared and this actually is equal to q over d squared 1 minus 1 over 1 plus d over d squared all right now we can write mclaren series mclaren series of 1 over 1 minus x squared why am i writing this well because the this part looks similar to that what is mclaren series of that all that is equal to summation from zero to infinity and times x to the n minus 1 all right all we need to do is to take this expression for e and then convert it into a series representation.

01:56 So we have e as q over d squared, q over d squared multiplied by 1 minus 1 over well.

02:09 Now, the important thing here is that we want this to be 1 minus x squared.

02:13 However, we have here 1 plus, right? so let's define our x in this scenario to be equal to negative of d over d.

02:22 So that we can write this one as 1 minus 1.

02:26 1 over 1 minus negative d over d squared.

02:32 And again, this is our x.

02:36 All right, this would then be written as q over d squared multiply by 1 minus...

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