{"product_id":"langmuira-langmuir-probe","title":"Langmuira – The Langmuir Probe \/ I ∝ exp(eV\/(kT)) Physics Sticker","description":"\u003cp\u003eLangmuira, a hooded plasma-reader, dips a thin wire probe into a glowing violet discharge.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eStick a small electrode into a plasma and sweep its voltage. Irving Langmuir and Harold Mott-Smith worked out what the current tells you (“The Theory of Collectors in Gaseous Discharges”, Phys. Rev. 28, 727, 1926, after their 1924 papers). Below the plasma potential the probe repels most electrons, and only those with enough energy to climb the barrier arrive. For a Maxwellian electron gas that electron current rises exponentially with probe voltage, so the slope of ln I gives the electron temperature. Above the plasma potential the current saturates.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eI\u003csub\u003ee\u003c\/sub\u003e ∝ exp(e(V − V\u003csub\u003ep\u003c\/sub\u003e)\/(kT\u003csub\u003ee\u003c\/sub\u003e))\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eI\u003csub\u003ee\u003c\/sub\u003e: electron current to the probe (A)\u003c\/li\u003e\n\u003cli\u003eV: probe voltage (V)\u003c\/li\u003e\n\u003cli\u003eV\u003csub\u003ep\u003c\/sub\u003e: plasma potential (V)\u003c\/li\u003e\n\u003cli\u003ek: Boltzmann constant (J\/K)\u003c\/li\u003e\n\u003cli\u003eT\u003csub\u003ee\u003c\/sub\u003e: electron temperature (K)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eLeft alone, a probe charges to the floating potential, below the plasma potential, where electron and ion currents cancel.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e The exponential holds only in the electron-retarding region and assumes a Maxwellian distribution and a thin sheath; magnetised or flowing plasmas need more care.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eLangmuira holds a thin probe whose tip curls into glowing violet loops, in front of a purple glowing tube. Printed text: \u003cstrong\u003eLANGMUIRA\u003c\/strong\u003e with the line \u003cstrong\u003eI ∝ exp(eV\/(kT))\u003c\/strong\u003e. Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964049231975,"sku":"21530177501352320752","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964049264743,"sku":"11310809970375050119","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964049297511,"sku":"12135111598366667138","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964049330279,"sku":"28672068939601763934","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/14523440095615733376_1200.jpg?v=1793658880","url":"https:\/\/pixelatedphysics.com\/id\/products\/langmuira-langmuir-probe","provider":"Pixelated Physics","version":"1.0","type":"link"}