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This shows the graphics and animation of a conductor that oscillates along the `x` direction and is braked by the the interaction of closed eddy current loops induced in it by a permanent magnet (PM) and the magnetic field of the magnet. The direction of the field of the PM is primarially along the `z` axis while the conductor lies in the `(x,y)` plane at a small adjustable distance `gap` from the North pole of the PM. The eddy currents' magnitudes are dependent on the variation `(dB_z)/dx`, in the `x` direction of the `z` component of the PM magnetic field as well as the speed, `v_x ` of the conductor in the `x` direction. As you can see from Canvas 1 after selecting Show `J`, the direction of the eddy current loops at the central region of the magnet is the opposite of that at larger x values beyond the the left and right edges of the magnet pole. Also, importantly, as you can see by selecting Show `B_z`, that component of the field of the PM is strong in the central region. For any eddy current charge, `q` moving at velocity, `bbv_q` , this gives rise to a Lorentz Force

`bbF=qbbv_qxxbbB`

which is opposite to the direction of motion of the conductor.As mentioned previously, the source of the eddy current density, `bbJ_E`, is mainly `(dB_z)/dx` since, near the conductor, `B_z(x)` is the only significant component of `bbB`. Faraday's law of induction for `bbJ_E` is

`bbgradxxbbJ_E(x,y)=-sigmabbhatz(dbbB)/(dx)(dx_C)/dt`

where `sigma` is the conductivity of the conductor and `(dx_C)/dt` is the speed of the conductor in the `x` direction.`intintgradbbF*dbbA=ointbbF.dbbl`

where the surface integral on left is over the area of a closed loop which and the line integral on the right is over the boundary of that loop. Using Stokes theorem to convert our differential equation to integral form we have`intint_("loop")bbgradxxbbJ_E(x,y)*dbbA=oint_("loop")bbJ_E.dbbl=-sigma(dx_C)/dtintint_("loop")(dbbB)/dxd*bbA`

From this equation we can get the average value of `bbJ_E` on any loop in the `(x-y)` plane if we know the average value of (dB_z)/dx going through the loop. So if we have a circular loop of radius `a` we can clearly say that the average value of the circulating `bbJ_E`.`bbJ_E=-sigma/(2pia) (dx_C)/dtintint_("loop")(dbbB)/dx*dbbA`

Note that the integral over `oint_("loop")bbJ_E.dbbl` will be largest when `bbJ_E` is circulating around the loop since then `bbJ_E` will be parallel with `bbdl`

`gradxxbbvecJ=|(bbhatx,bbhaty,bbhatz),(del/(delx),del/(dely),del/(delz)),(J_x,J_y,J_z)|` `=((delJ_z)/(dely)-(delJ_y)/(delz))bbhatx-((delJ_z)/(delx)-(delJ_x)/(dely))bbhaty+ ((delJ_y)/(delx)-(delJ_x)/(dely))bbhatz`

Since the right hand side of the expresssion for `bbJ` has only a `bbhatz` component, an expression for `bbJ` becomes:`(delJ_y)/(delx)-(delJ_x)/(dely)=-1/sigma(dB_z)/dx(dx_C)/dt`

Let's rename the expression of the right hand side of this equation`S(x_s,y_s)=-1/sigma(dB_z(x_s,y_s'))/dx(dx_C)/dt`

where `S` stands for source. The values of `J_y(x,y)` is given by integration of `S(x-x_s,y-y_s)` over all value of `(x_s,y_s)`. Let us assume that the dimension of the magnet along the `bbhaty` direction is very long which is the same as saying that we have very small `y` or `y_s` variation of `S(x-x_s,y-y_s)` In general `S(x-x_s,y-y_s)` has significant variation over `x_s` but very little variation over `y_s`. Then the integral`J_x(x,y)=-int_(-oo)^yS(x-x_s,y-y_s)dy_s`

results in a much smaller value than the similar integral:`J_y(x,y)=-int_(=oo)^xS(x-x_s,y-y_s)dx_s`

We choose to drop `J_x` term from(`J_x,J_y`) in the curl equation.If we press tje "Show `v_x(dB_z)/dx`" button we see that the variation on the left side starts positive and recedes going to the left. Similarly the variation starts very negative and recedes going to the right. Then the integral for `J_y(x,y)` can be written:

`J_y(x,y)=-int_oo^x_sS(x-x_s,y-y_s)dx_s`

and this result is shown when when the button "Show Vertical Current Density 'J_y' is pressed. "The only control provided is the gap between the magne and the conductor. Increasing the gap will decrease the maximum value of `bbB_z` at the conductor.

I show the directions and magnitude of the eddy currents, `J_y(x,y)`, induced in the conductor when it moves relative to the magnet. I also show that this eddy current density which derives from the source

`S(x-x_s,y-y_x)=(dB_z(x_s,y_s))/dx(dx_C)/dt`

reverses when the direction of the conductor motion is reversed. To simulate the current density motion in the animations I show charges, `q(x,y)`, moving in clockwise or counter clockwise loops around the magnet edges. Then, when the conductor motion direction reverses, the eddy charge loops' direction also reverses. In addition, the value of the charges near the center of the magnet pole are more concentrated than those to the left and right of the pole so that the product integral`int_-oo^ooB_z(x)J_E(x)dx`

which is proportional to the average force on the conductor would be large. Note that since `intB_z(x)dB_z(x)=(B_z^2(x))/2` that this integral is easy`(dx_C)/dtint_-oo^ooB_z(x)(dB_z)/dxdx=(dx_C)/dt(B_z^2(x))/2]_(x=-oo)^(x=oo)`

so that if `B_z(x)` were symmetrical, then the integral is 0.Canvas 2:Top View of Eddy Current Density `bbJ_y(x,y)` at Conductor Surface

Canvas 2:Top View of Conductor Speed, `v_x`, Product with `bbB` Field Differential, `v_x(dbbB_z)/dx`, of PM at Conductor Surface.

Canvas 2:End View of Field `bbB_z` of PM at Conductor Surface