Charged Ring

This article will describe the Electric Field created by a uniformly charged thin ring and how to calculate the value of this field.

The Main Idea

Objects that create electric fields come in various shapes, including circular rings. Rings are

A Mathematical Model

What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ {\frac{d\vec{p}}{dt}}_{system} = \vec{F}_{net} }[/math] where p is the momentum of the system and F is the net force from the surroundings.

A Computational Model

Below is a representation of adding up the electric fields created by individual pieces of a ring to deduce its electric field.

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(function() {

var __g=typeof global!=='undefined'?global:window;__g=(__g.__streamline||(__g.__streamline={}));__g.setEF=__g.setEF||function(e,f){e.__frame = e.__frame||f};var __srcName='undefined_.js'; function __func(_,__this,__arguments,fn,index,frame,body){if(!_){return __future.call(__this,fn,__arguments,index)}frame.file=__srcName;frame.prev=__g.frame;__g.frame=frame;try{body()}catch(e){__g.setEF(e,frame.prev);__propagate(_,e)}finally{__g.frame=frame.prev}} function __cb(_,frame,offset,col,fn){frame.offset=offset;frame.col=col;var ctx=__g.context;return function ___(err,result){var oldFrame=__g.frame;__g.frame=frame;__g.context=ctx;try{if(err){__g.setEF(err,frame);return _(err)}return fn(null,result)}catch(ex){__g.setEF(ex,frame);return __propagate(_,ex)}finally{__g.frame=oldFrame}}} function __future(fn,args,i){var done,err,result;var cb=function(e,r){done=true;err=e,result=r};args=Array.prototype.slice.call(args);args[i]=function ___(e,r){cb(e,r)};fn.apply(this,args);return function ___(_){if(done)_.call(this,err,result);else cb=_.bind(this)}.bind(this)} function __propagate(_,err){try{_(err)}catch(ex){__trap(ex)}} function __trap(err){if(err){if(__g.context&&__g.context.errorHandler)__g.context.errorHandler(err);else console.error("UNCAUGHT EXCEPTION: "+err.message+"\n"+err.stack)}} /* 1 */ function main(wait) {

             var version, box, sphere, cylinder, pyramid, cone, helix, ellipsoid, ring, arrow, graph, display, vector, _$GS_$END, scene, _$rapyd$_Temp, _$rapyd$_print, arange, s, a, obsloc, Eb, dtheta, angles, b, pt, ra, Ealist, Ea, r, theta, _$rapyd$_Iter0, _$rapyd$_Index0, _$rapyd$_Iter1, _$rapyd$_Index1;

/* 43 */ function axes(a) { /* 44 */ if ((typeof a === "undefined")) { /* 44 */ a = 10;

               };

/* 45 */ var axes; /* 47 */ axes = curve({ /* 47 */ pos: [vector(a["-u"](), 0, 0),vector(a, 0, 0),vector(0, 0, 0),vector(0, a["-u"](), 0),vector(0, a, 0),vector(0, 0, 0),vector(0, 0, a["-u"]()),vector(0, 0, a),], /* 47 */ color: color.black

               });
             };
             var __frame = {
               name: "main",
               line: 1
             };
             return __func(wait, this, arguments, main, 0, __frame, function __$main() {

/* 3 */ version = ["1.1","glowscript",]; /* 4 */ window.__GSlang = "vpython"; /* 5 */ box = vp_box; /* 6 */ sphere = vp_sphere; /* 7 */ cylinder = vp_cylinder; /* 8 */ pyramid = vp_pyramid; /* 9 */ cone = vp_cone; /* 10 */ helix = vp_helix; /* 11 */ ellipsoid = vp_ellipsoid; /* 12 */ ring = vp_ring; /* 13 */ arrow = vp_arrow; /* 14 */ graph = vp_graph; /* 15 */ display = canvas; /* 16 */ vector = vec; /* 17 */ _$GS_$END = 0; /* 18 */ scene = canvas(); /* 19 */ _$rapyd$_Temp = 0; /* 20 */ _$rapyd$_print = GSprint; /* 21 */ arange = range; /* 23 */ scene.background = color.white; /* 25 */ scene.width = 1000; /* 27 */ scene.height = 700; /* 29 */ scene.range = 5; /* 31 */ scene.forward = vector(0.2["-u"](), 0.1["-u"](), 1["-u"]()); /* 33 */ scene.center = vector(4, 0, 0); /* 35 */ s = "Calculate E of ring by superposition of E of point charges.\n"; /* 37 */ s += "Magenta arrow is deltaE from this segment; orange arrow is net field so far.\n"; /* 39 */ s += "Click to advance. Zoom out or rotate when E gets large."; /* 41 */ label({ /* 41 */ pos: vector(4, 4.3["-u"](), 0), /* 41 */ text: s, /* 41 */ color: color.black, /* 41 */ box: 0, /* 41 */ opacity: 1

               });

/* 50 */ axes(4); /* 52 */ a = ring({ /* 52 */ pos: vector(0, 0, 0), /* 52 */ radius: 3, /* 52 */ color: color.red, /* 52 */ thickness: 0.05

               });

/* 54 */ obsloc = vector(2, 0, 0); /* 56 */ Eb = arrow({ /* 56 */ pos: obsloc, /* 56 */ axis: vector(0, 0, 0), /* 56 */ shaftwidth: 0.1, /* 56 */ color: color.orange

               });

/* 58 */ dtheta = pi["/"](10); /* 60 */ angles = arange(0, 2["*"](pi), dtheta); /* 62 */ b = 25; /* 64 */ return scene.waitfor("click", __cb(wait, __frame, 63, -1, function __$main() { /* 66 */ pt = sphere({ /* 66 */ pos: vector(0, a.radius, 0), /* 66 */ color: color.cyan, /* 66 */ radius: a.thickness["*"](1.2)

                 });

/* 68 */ ra = arrow({ /* 68 */ pos: pt.pos, /* 68 */ axis: vector(0, 0, 0), /* 68 */ color: color.green, /* 68 */ shaftwidth: 0.05, /* 68 */ fixedwidth: 1

                 });

/* 70 */ Ealist = []; /* 72 */ _$rapyd$_Iter0 = angles; /* 73 */ _$rapyd$_Index0 = 0;

                 var __3 = false;
                 return (function ___(__break) {
                   var __more;
                   var __loop = __cb(wait, __frame, 0, 0, function __$main() {
                     __more = false;
                     if (__3) {

/* 73 */ _$rapyd$_Index0++;

                     }
                      else {
                       __3 = true;
                     }
                   ;

/* 73 */ var __2 = (_$rapyd$_Index0 < _$rapyd$_Iter0.length);

                     if (__2) {

/* 74 */ theta = _$rapyd$_Iter0[_$rapyd$_Index0]; /* 76 */ Ea = arrow({ /* 76 */ pos: obsloc, /* 76 */ axis: vector(0, 0, 0), /* 76 */ shaftwidth: 0.04, /* 76 */ color: color.magenta

                       });

/* 78 */ Ealist.append(Ea); /* 80 */ pt.pos = vector(0, a.radius["*"](cos(theta)), a.radius["*"](sin(theta))); /* 82 */ ra.pos = pt.pos; /* 84 */ r = Ea.pos["-"](ra.pos); /* 86 */ ra.axis = r; /* 88 */ Ea.axis = b["/"](Math.pow(mag(r), 2))["*"](norm(r)); /* 90 */ Eb.axis = Eb.axis["+"](Ea.axis); /* 92 */ return scene.waitfor("click", __cb(wait, __frame, 91, -1, function __$main() {

                         while (__more) {
                           __loop();
                         };
                         __more = true;
                       }));
                     }
                      else {
                       __break();
                     }
                   ;
                   });
                   do {
                     __loop();
                   } while (__more);
                   __more = true;
                 })(function __$main() {

/* 95 */ return scene.waitfor("click", __cb(wait, __frame, 94, -1, function __$main() { /* 97 */ ra.visible = false; /* 99 */ pt.visible = false; /* 101 */ _$rapyd$_Iter1 = Ealist; /* 102 */ for (_$rapyd$_Index1 = 0; (_$rapyd$_Index1 < _$rapyd$_Iter1.length); _$rapyd$_Index1++) { /* 103 */ Ea = _$rapyd$_Iter1[_$rapyd$_Index1]; /* 105 */ Ea.visible = false;

                     };
                     wait();
                   }));
                 });
               }));
             });
           };

/* 108 */ main;

$(function(){ window.__context = { glowscript_container

$("#glowscript").removeAttr("id") }; main() })})()

</script>

Examples

Below are examples of calculating the value and direction of the Electric Field caused by a ring positioned in the xy-plane with a radius R and total charge q.

Connectedness

Calculating the electric field of a charged ring may seem like another exercise to practice visualizing the way a collection of charge can create a field and influence other objects around it, but it becomes very important in the study of Simple Circuits.

Electric fields created by rings of charge are the source of currents in circuits, which makes these little charged rings one of the most important components of daily life for almost all Americans and 83% of the citizens of the world. Rings of charge compounded into hollow wires of charge are a central part of past, present, and future human innovation.

History

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

See also

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

Further reading

Books, Articles or other print media on this topic

External links

Here are some resources for further reading if you're interested:

[www.website.com Website]

References

"Electric Field on the Axis of a Ring of Charge". University of Delaware Physics Library. Adapted from Stephen Kevan's lecture on Electric Fields and Charge Distribution. April 8, 1996. http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html

Chabay, R., & Sherwood, B. (2015). Matter and Interactions (4th ed., Vol. 2, pp. 597-599). Wiley.

Energy Access Database. (n.d.). Retrieved December 6, 2015, from http://www.worldenergyoutlook.org/resources/energydevelopment/energyaccessdatabase/

Work in progress! - afrancis38