Magnetic Field of a current distribution

In summary, a magnetic lens can be created by placing four long, thin current carrying conducting sheets on the sides of a square. The currents in the sheets are distributed uniformly and directed either into or out of the plane of the page. The values of ##H_{x}## and ##H_{y}## can be determined at various points, including the boundary just to the left and below the current sheet, and at the origin. In the interior region, the equations for ##H_{x}## and ##H_{y}## can be derived and the equation for the field lines is found to be dependent on the tangent hyperbolic function. In the case of a beam of positively charged particles being injected into the interior region
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Homework Statement



A magnetic lens is made by placing four long, thin current carrying conducting sheets of width ##2a## on the sides of a square, as shown in the figure. The currents in the conducting sheets are distributed uniformly over the sheets, and are directed either into or out of the plane of the page, as shown in the figure. You may neglect the effects of ends and corners.

Capture.jpg


(a) What are the values of ##H_{x}(a^{-},y)## and ##H_{y}(a^{-},y)## along the boundary just to the left of the current sheet at ##x=a##? Likewise, what are the values of ##H_{x}(x,a^{-})## and ##H_{y}(x,a^{-})## along the upper boundary just below the current sheet at ##y=a##?

(b) What are ##H_{x}## and ##H_{y}## at the origin?

(c) Find ##H_{x}## and ##H_{y}## in the interior region ##-a<x<a## and ##-a<y<a##?

(d) Derive the equation which describes the field lines in the interior region.

(e) Suppose that a beam of positively charged particles is injected into the interior region of the lens so that their velocities are initially along the ##z##-axis (out of the page). Discuss how the lens can act to focus the beam. Try to be as quantitative as possible.

Homework Equations



The Attempt at a Solution



(a) ##H_{x}(a^{-},y) = 0## and ##H_{y}(a^{-},y) = \frac{\mu_{0}I}{2\pi a^{-}}##.

##H_{x}(x,a^{-}) = \frac{\mu_{0}I}{2\pi a^{-}}## and ##H_{y}(x,a^{-}) = 0##.

Please check my answers to (a).
 
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(b) ##H_{x}(0,0) = 0## and ##H_{y}(0,0) = 0##.(c) ##H_{x} = \frac{\mu_{0}I}{2\pi a^{-}}[tanh(\frac{y}{a}) - tanh(\frac{x}{a})]## and ##H_{y} = -\frac{\mu_{0}I}{2\pi a^{-}}[tanh(\frac{x}{a}) + tanh(\frac{y}{a})]##.(d) The equation which describes the field lines in the interior region is given by: ##\frac{dy}{dx} = -\frac{H_{x}(x,y)}{H_{y}(x,y)} = \frac{tanh(\frac{x}{a})+tanh(\frac{y}{a})}{tanh(\frac{y}{a})-tanh(\frac{x}{a})}##.(e) The lens will act to focus the beam by causing the particles to be deflected towards the center of the lens. This can be quantified by noting that the force experienced by the particles is given by ##F=qv\times H##, where ##v## is the velocity of the particles, and ##H## is the magnetic field produced by the lens. If the particles are initially moving along the ##z##-axis, then they will experience a force in the ##x##- and ##y##-directions, which will cause them to be deflected towards the center of the lens. The amount of deflection will depend on the magnitude of the magnetic field, and thus on the magnitude of the current in the conducting sheets.
 

1. What is a magnetic field of a current distribution?

A magnetic field of a current distribution refers to the magnetic field that is created by a flow of electric current through a material or space.

2. How is the magnetic field of a current distribution calculated?

The magnetic field of a current distribution can be calculated using the Biot-Savart law, which states that the strength of the magnetic field at a point is directly proportional to the current, the distance from the point to the current, and the sine of the angle between the current and the line connecting the point and the current.

3. What factors affect the strength of the magnetic field of a current distribution?

The strength of the magnetic field of a current distribution is affected by the magnitude of the current, the distance from the current, and the orientation of the current with respect to the point of measurement.

4. How is the direction of the magnetic field of a current distribution determined?

The direction of the magnetic field of a current distribution is determined by the right-hand rule, which states that if the thumb of the right hand points in the direction of the current, then the fingers curl in the direction of the magnetic field.

5. What are some real-life applications of the magnetic field of a current distribution?

The magnetic field of a current distribution has many practical applications, such as in motors and generators, transformers, MRI machines, and particle accelerators. It is also used in compasses, magnetic levitation trains, and magnetic field sensors.

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