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mysearch
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Main Question or Discussion Point
If possible, I would like to confirm or correct my assumptions associated with the model described below and illustrated in the attached diagram. Most descriptions tend to aggregate effects of moving charge as a current flowing in a wire, but the specific purpose of this model is to try to examine the same concept in the isolation of 2 charges.
The model, subject to the Lorentz transforms, is shown in both a stationary and moving reference frame. It consists of a positive charge [Q] moving with velocity [v1] and a negative charge [q] moving parallel to [Q] along the xaxis with velocity [v2]. For the purpose of the model [v1=v2=v], but the subscript is only used to identify the relevance in specific equations cited below.
With reference to the diagram, the model on the left is linked to the stationary frame labelled [O*], while the model on the right is the moving frame labelled [O]. In the stationary [O*] frame there is no Bfield and the net force is directly related to the Efield, which points outwards due to the sign of the charge [Q]. In the moving [O] frame, a Bfield does exist in the direction shown, i.e. perpendicular in zaxis, and has an associated force that acts inwards, i.e. yaxis, in opposition to the outward Eforce [Fe].
[1] [tex] E= \frac {Q}{4 \pi \epsilon_0 y^2}* \frac {1 \beta^2}{(1 \beta^2 sin^2 \theta)^{3/2}} [/tex]
[2] [tex] B= \frac {Qv_1}{4 \pi \epsilon_0 c^2 y^2}* \frac {1 \beta^2}{(1 \beta^2 sin^2 \theta)^{3/2}} [/tex]
[3] [tex] B = \frac {vE}{c^2}[/tex]; where [tex] sin \theta =1[/tex]
While the form of [1] and [2] is orientated towards the moving frame, the model restricts [r] to the yaxis, which is not subject to length contraction, i.e. y=y*, and therefore the values for the stationary frame can be determined by setting [v=0]. As such, the second term in [1] and [2] disappears and we are left with the standard forms of (E) and (B). However, the corresponding force equations are given below:.
[4] [tex]F = qE + qv_2Bsin \theta [/tex]
[5] [tex]F_e =qE; F_b = qv_2B [/tex]
The form of [4] only implies the magnitude of the force, not the direction, which is addressed in the diagram. Again, [5] reduces [4] to the components acting along the yaxis such that [tex] \theta=90[/tex]. However, it is highlighted that [3] and [5] have cited different velocities [v1] and [v2] in order to clarify an issue.
Conceptually, it seems that the Bfield is thought to exist based on [Q] being in motion with velocity [v1]. However, in practice, the existence of a force (Fb) associated with the field can only be measured by a secondary test charge [q] moving through(?) the Bfield with velocity [v2]. However, in this case, [Q] and [q] have the same velocity, so is there any need for the velocity of [q] to be relative to the Bfield?
The attached diagram assumes that [v1] and [v2] can both be substituted for [v] in [3] and [5] and, as such, the diagram shows the moving frame having a magnetic force (Fb) that opposes (Fe). However, when [v<<c], equation [3][5] can be combined and reduced to the following form, again noting the restriction to the yaxis.
[6] [tex] F_e = \frac {Qq}{4 \pi \epsilon_0 y^2} [/tex]
[7] [tex] F_b = \frac {Qq}{4 \pi \epsilon_0 y^2} * \frac {v^2}{c^2} [/tex]
Comparing [6] and [7], when (v<<c), suggests that (Fb) may be 1617 orders of magnitude smaller than (Fe), such that the net force will still repel [q] away from [Q]. Anyway, I would appreciate any correction to the model, where necessary. Thanks.
The model, subject to the Lorentz transforms, is shown in both a stationary and moving reference frame. It consists of a positive charge [Q] moving with velocity [v1] and a negative charge [q] moving parallel to [Q] along the xaxis with velocity [v2]. For the purpose of the model [v1=v2=v], but the subscript is only used to identify the relevance in specific equations cited below.
With reference to the diagram, the model on the left is linked to the stationary frame labelled [O*], while the model on the right is the moving frame labelled [O]. In the stationary [O*] frame there is no Bfield and the net force is directly related to the Efield, which points outwards due to the sign of the charge [Q]. In the moving [O] frame, a Bfield does exist in the direction shown, i.e. perpendicular in zaxis, and has an associated force that acts inwards, i.e. yaxis, in opposition to the outward Eforce [Fe].
[1] [tex] E= \frac {Q}{4 \pi \epsilon_0 y^2}* \frac {1 \beta^2}{(1 \beta^2 sin^2 \theta)^{3/2}} [/tex]
[2] [tex] B= \frac {Qv_1}{4 \pi \epsilon_0 c^2 y^2}* \frac {1 \beta^2}{(1 \beta^2 sin^2 \theta)^{3/2}} [/tex]
[3] [tex] B = \frac {vE}{c^2}[/tex]; where [tex] sin \theta =1[/tex]
While the form of [1] and [2] is orientated towards the moving frame, the model restricts [r] to the yaxis, which is not subject to length contraction, i.e. y=y*, and therefore the values for the stationary frame can be determined by setting [v=0]. As such, the second term in [1] and [2] disappears and we are left with the standard forms of (E) and (B). However, the corresponding force equations are given below:.
[4] [tex]F = qE + qv_2Bsin \theta [/tex]
[5] [tex]F_e =qE; F_b = qv_2B [/tex]
The form of [4] only implies the magnitude of the force, not the direction, which is addressed in the diagram. Again, [5] reduces [4] to the components acting along the yaxis such that [tex] \theta=90[/tex]. However, it is highlighted that [3] and [5] have cited different velocities [v1] and [v2] in order to clarify an issue.
Conceptually, it seems that the Bfield is thought to exist based on [Q] being in motion with velocity [v1]. However, in practice, the existence of a force (Fb) associated with the field can only be measured by a secondary test charge [q] moving through(?) the Bfield with velocity [v2]. However, in this case, [Q] and [q] have the same velocity, so is there any need for the velocity of [q] to be relative to the Bfield?
The attached diagram assumes that [v1] and [v2] can both be substituted for [v] in [3] and [5] and, as such, the diagram shows the moving frame having a magnetic force (Fb) that opposes (Fe). However, when [v<<c], equation [3][5] can be combined and reduced to the following form, again noting the restriction to the yaxis.
[6] [tex] F_e = \frac {Qq}{4 \pi \epsilon_0 y^2} [/tex]
[7] [tex] F_b = \frac {Qq}{4 \pi \epsilon_0 y^2} * \frac {v^2}{c^2} [/tex]
Comparing [6] and [7], when (v<<c), suggests that (Fb) may be 1617 orders of magnitude smaller than (Fe), such that the net force will still repel [q] away from [Q]. Anyway, I would appreciate any correction to the model, where necessary. Thanks.
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