{"id":2533,"date":"2024-08-24T08:22:47","date_gmt":"2024-08-24T08:22:47","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=2533"},"modified":"2024-08-24T08:22:47","modified_gmt":"2024-08-24T08:22:47","slug":"temperature-co-efficient-of-resistance","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/08\/24\/temperature-co-efficient-of-resistance\/","title":{"rendered":"TEMPERATURE CO-EFFICIENT OF RESISTANCE"},"content":{"rendered":"\n

Consider a metallic resistor having a resistance of R<\/em>0<\/sub> and 0\u00b0C and R<\/em>t<\/sub> at t<\/em>\u00b0C. The increase in resistance (R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) is directly proportional to its initial resistance, that is,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

(R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) \u221d R<\/em>0<\/sub><\/p>\n\n\n\n

(R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) is directly proportional to rise in temperature, that is<\/p>\n\n\n\n

 <\/p>\n\n\n\n

(R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) \u221d t<\/em><\/p>\n\n\n\n

 (R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) depends on the nature of its material.<\/p>\n\n\n\n

Thus,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

(R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) \u221d R<\/em>0<\/sub>t<\/em><\/p>\n\n\n\n

 or<\/p>\n\n\n\n

 <\/p>\n\n\n\n

(R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>) = \u03b1<\/em>0<\/sub>R<\/em>0<\/sub>t<\/em>        (1.1)<\/p>\n\n\n\n

 <\/p>\n\n\n\n

where \u03b1<\/em>0<\/sub> is a constant called temperature coefficient of resistance at 0\u00b0C. Its value depends upon the nature of resistor material.<\/p>\n\n\n\n

By rearranging the\u00a0equation (1.1), we get<\/p>\n\n\n\n

 <\/p>\n\n\n\n

R<\/em>t<\/sub> <\/em>= R<\/em>0<\/sub> (1 + \u03b1<\/em>0<\/sub>t<\/em>)        (1.2)<\/p>\n\n\n\n

 <\/p>\n\n\n\n

and<\/p>\n\n\n\n

\"img\"        (1.3)<\/p>\n\n\n\n

If R<\/em>0<\/sub> = 1 ohm; t<\/em> = 1\u00b0C; then, \u03b1<\/em>0<\/sub> = (R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>)<\/p>\n\n\n\n

Hence, temperature coefficient of resistance at 0\u00b0C may be defined as the change in resistance per ohm original resistance per \u00b0C change in temperature.<\/p>\n\n\n\n

Unit: <\/strong>We know that, \u03b1<\/em>0<\/sub> = (R<\/em>t<\/sub><\/em> \u2212 R<\/em>0<\/sub>)\/R<\/em>0 <\/sub>t<\/em><\/p>\n\n\n\n

<\/a>Substituting the units of various quantities, we get,<\/p>\n\n\n\n

Unit of \"img\"<\/p>\n\n\n\n

Hence, the unit of temperature coefficient is per \u00b0C.<\/p>\n\n\n\n

1.20  TEMPERATURE CO-EFFICIENT OF COPPER AT 0\u00b0C<\/h5>\n\n\n\n

It has been seen that R<\/em>t<\/sub><\/em> = R<\/em>0<\/sub> (1 + \u03b1<\/em>0<\/sub>t<\/em>)<\/p>\n\n\n\n

The above equation holds good for both rise and fall in temperature. The temperature verses resistance graph of copper material is a straight line as shown in Figure 1.9<\/a>.<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Fig. 1.9  <\/strong>Graph between temperature and resistance<\/p>\n\n\n\n

If this line is extended in the backward direction, it would cut the temperature axis at point A where temperature is \u2212234.5\u00b0C.<\/p>\n\n\n\n

Putting the value of R<\/em>t<\/sub><\/em> = 0 and t<\/em> = \u2212234.5\u00b0C in the above equation, we get,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

0 = R<\/em>0<\/sub> [1 + \u03b1<\/em>0<\/sub> (\u2212234.5)]<\/p>\n\n\n\n

or<\/p>\n\n\n\n

 <\/p>\n\n\n\n

234.5 \u03b1<\/em>0<\/sub> = 1<\/p>\n\n\n\n

or<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

where \u03b1<\/em>0<\/sub> is the temperature coefficient of resistance of copper at 0\u00b0C.<\/p>\n\n\n\n

However, in practice, the curve departs (at point B) from a straight line at very low temperature and the resistance never becomes zero.<\/p>\n\n\n\n

1.21  EFFECT OF TEMPERATURE ON \u03b1<\/em><\/h5>\n\n\n\n

The value of temperature coefficient of resistance (\u03b1<\/em> ) is not constant. Its value depends upon the initial temperature on which the increment in resistance is based. If the initial temperature is 0\u00b0C, the value of \u03b1<\/em> is \u03b1<\/em>0<\/sub>. Similarly, if the initial temperature is t<\/em>1<\/sub>\u00b0C, the value of \u03b1<\/em> is \u03b1<\/em>1<\/sub>.<\/p>\n\n\n\n

Relation between <\/em>\u03b1<\/em>0<\/sub> and \u03b1<\/em>1<\/sub>.<\/p>\n\n\n\n

Consider a conductor of resistance R<\/em>0<\/sub> at 0\u00b0C. When its temperature is raised to t<\/em>1<\/sub>\u00b0C, <\/em>its resistance increases to say R<\/em>1<\/sub>.<\/p>\n\n\n\n

\u2234<\/p>\n\n\n\n

 <\/p>\n\n\n\n

(R<\/em>1<\/sub> \u2212 R<\/em>0<\/sub>) = R<\/em>0<\/sub>\u03b1<\/em>0<\/sub>t<\/em>1        <\/sub>(1.4)<\/p>\n\n\n\n

or<\/p>\n\n\n\n

 <\/p>\n\n\n\n

R<\/em>1<\/sub> = R<\/em>0<\/sub> (1 + \u03b1<\/em>0<\/sub>t<\/em>1<\/sub>)        (1.5)<\/p>\n\n\n\n

Let us suppose that the conductor of resistance R<\/em>1<\/sub> at t<\/em>1<\/sub>\u00b0C be now cooled down to 0\u00b0C to give a resistance of final value R<\/em>0<\/sub>.<\/p>\n\n\n\n

<\/a>Then,<\/p>\n\n\n\n

 <\/p>\n\n\n\n

R<\/em>0<\/sub> = R<\/em>1<\/sub> (1 + \u03b1<\/em>1<\/sub> (\u2212t<\/em>1<\/sub>)) = R<\/em>1<\/sub> \u2212 t<\/em>1<\/sub>R<\/em>1<\/sub>\u03b1<\/em>1<\/sub><\/p>\n\n\n\n

or<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Substituting the value of (R<\/em>1<\/sub> \u2212 R<\/em>0<\/sub>) from equation (1.4)<\/a> and the value of R<\/em>1<\/sub> from equation (1.5)<\/a>, we get,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

or<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Relation between \u03b1<\/em>1 <\/sub>and \u03b1<\/em>2<\/sub>.<\/p>\n\n\n\n

Rearranging the equation (1.6)<\/a>, we get,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Similarly, if the initial temperature is t<\/em>2<\/sub>\u00b0C and, the value of \u03b1<\/em> is \u03b1<\/em>2<\/sub>, then<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

Subtracting equation (1.7)<\/a> from (1.8)<\/a>, we get,<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

or<\/p>\n\n\n\n

\"img\"\/<\/figure>\n\n\n\n

The following conclusions were drawn from the discussion<\/p>\n\n\n\n

    \n
  1. From\u00a0equation (1.10), it is clear that with the rise in temperature, the value of\u00a0\u03b1<\/em>\u00a0decreases. Thus,\u00a0\u03b1<\/em>0<\/sub>\u00a0has the maximum value.<\/li>\n\n\n\n
  2. If the initial resistance at 0\u00b0C is R<\/em>0<\/sub> and the final resistance at t<\/em>1<\/sub> \u00b0C is R<\/em>1<\/sub>, then R<\/em>1<\/sub> = R<\/em>0<\/sub> (1 + \u03b1<\/em>0<\/sub>t<\/em>1<\/sub>) <\/li>\n\n\n\n
  3. If the initial resistance at t<\/em>1<\/sub> \u00b0C is R<\/em>1<\/sub> and the final resistance at t<\/em>2 <\/sub>\u00b0C is R<\/em>2<\/sub>, then R<\/em>2<\/sub> = R<\/em>1<\/sub> (1 + \u03b1<\/em>1<\/sub> (t<\/em>2<\/sub> \u2212 t<\/em>1<\/sub>)) <\/li>\n\n\n\n
  4. If the value of \u03b1<\/em> at t<\/em>1<\/sub>\u00b0C is \u03b1<\/em>1<\/sub>, then its value at t<\/em>2<\/sub>\u00b0C will be \"img\"<\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Consider a metallic resistor having a resistance of R0 and 0\u00b0C and Rt at t\u00b0C. The increase in resistance (Rt \u2212 R0) is directly proportional to its initial resistance, that is,   (Rt \u2212 R0) \u221d R0 (Rt \u2212 R0) is directly proportional to rise in temperature, that is   (Rt \u2212 R0) \u221d t  (Rt \u2212 R0) depends on the nature of its material. Thus,   (Rt \u2212 R0) \u221d R0t  or   (Rt \u2212 R0) […]<\/p>\n","protected":false},"author":1,"featured_media":2504,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[403],"tags":[],"class_list":["post-2533","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-concepts-of-circuit-theory"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/08\/lightning.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/2533","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=2533"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/2533\/revisions"}],"predecessor-version":[{"id":2534,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/2533\/revisions\/2534"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/2504"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=2533"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=2533"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=2533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}