Field of a Moving Charge

A charged particle with charge q₁ moves with constant velocity v along the z-axis (see Figure 1.1). Its potentials are

Where s = [(1-β²) (x²-y²) + (x-vt)²]^1/2 and β = v/c

Figure 1.1

a) Show that ϕ and A satisfy the Lorentz condition

b) Calculate the fields E and B at point P (x, y, z) and time t. Show first that B = (v/c) × E, and calculate E explicitly. Show that E is parallel to n.

c) Assume at P a second particle with charge q₂ moving with the same velocity v as the first. Calculate the force on q₂.

SOLUTION

a) Differentiating, we obtain

(1)

and, since v lies only in the z direction (see Figure 1.2)

(2)

b) To calculate B, we recall from the definition of A that

(3)

where we have used the fact that

Now,

(4)

Figure 1.2

where  is the vector from the charge to point P and therefore is parallel to n. The same results can be obtained by calculating E in the moving frame (with the charge q₁) and then taking the Lorentz transformation.

c) The force acting on charge q₂ can be calculated as the force acting on q₂ in the field of q₁ from (4):

(5)

where we used (3). Substituting E from (4), we find

We can express through the angle θ between R and ẑ (see Figure 1.2):

where we used x² + y² = R² sin θ, and so s = |R| (1- β² sin² θ)^1/2. Now, F may be written as a sum of the projections perpendicular and parallel to the z direction:

مواضيع ذات صلة

مصطلحات الفيزياء النووية

إزاحة حمراء REDSHIFT

المركبات النانوية (Nanocomposites)

مولد فان دي جراف الالكتروستاتي

التطويرات التقنية الحديثة في المعجلات الالكتروستاتية

معجلات الالكترونات الالكتروستاتية

معجلات الفان دي جراف الترادفية

الحد من زيادة الطاقة في معجل فان دي جراف

أنبوبة التعجيل قي معجلات الجسيمات من نوع فان دي جراف

ارتكاز معجل فان دي جراف

مبدأ تشغيل معجل الفان دي جراف

القضايا الأخلاقية المتعلقة بتكنولوجيا النانو