Finding the B-field at a point outside ring of current IN Plane of ring

[maxmax1]

Homework Statement

Determine the B field at a point P with distance d from centre of current ring radius r (d>r ie outside ring of current) but IN the Plane of ring (i.e off the axis)

Homework Equations

Biot savart: db=(u.I/4pi.r^2).dl.sinx

u=mu, I=current, x=angle made with current vector element dl and vector r connecting P to dl.

The Attempt at a Solution

the angle x will vary at all points along ring with vector joining point P, as will r.

I will integrate from 0 to 2pi my expression for db but with r^2=a^2+d^2-2adcosy

where y is angle between radial vector and d.

I am stuck because cannot determine how to express y in terms of x. and thus cannot integrate!

many thanks!

 

[tiny-tim]

I am stuck because cannot determine how to express y in terms of x. and thus cannot integrate!

Hi maxmax1!

Hint: P is at (d,0).

The tangent at Q = (r cosy, r siny) is … ?

So the angle betweent he tangent and PQ is … ?

 

[baba89]

Your approach to using the Biot-Savart law to find the B-field at a point outside the ring of current in the plane of the ring is correct. However, you are correct in noting that the angle y between the radial vector and d will vary at different points along the ring, making it difficult to integrate. In this case, it may be helpful to use vector calculus to solve for the B-field.

One approach could be to define a vector function for the current element dl and the vector r connecting P to dl, and then use the cross product of these two vectors to find the direction of the B-field at point P. From there, you can integrate over the entire ring to find the total B-field at point P.

Another approach could be to use the concept of symmetry and break the ring into smaller elements that are easier to integrate. For example, you could consider dividing the ring into quarters and integrating over each quarter separately, taking into account the different angles and distances at each point.

Overall, finding the B-field at a point outside the ring of current in the plane of the ring can be a complex problem, but with careful consideration and the use of vector calculus, it is possible to find a solution.