{"id":299070,"date":"2024-08-17T11:13:54","date_gmt":"2024-08-17T11:13:54","guid":{"rendered":"https:\/\/www.electricity-magnetism.org\/?p=299070"},"modified":"2024-09-08T11:22:20","modified_gmt":"2024-09-08T11:22:20","slug":"%e0%a4%b8%e0%a4%ae%e0%a4%be%e0%a4%a8-%e0%a4%b0%e0%a5%82%e0%a4%aa-%e0%a4%b8%e0%a5%87-%e0%a4%86%e0%a4%b5%e0%a5%87%e0%a4%b6%e0%a4%bf%e0%a4%a4-%e0%a4%a1%e0%a4%bf%e0%a4%b8%e0%a5%8d%e0%a4%95-%e0%a4%95","status":"publish","type":"post","link":"https:\/\/www.electricity-magnetism.org\/hi\/%e0%a4%b8%e0%a4%ae%e0%a4%be%e0%a4%a8-%e0%a4%b0%e0%a5%82%e0%a4%aa-%e0%a4%b8%e0%a5%87-%e0%a4%86%e0%a4%b5%e0%a5%87%e0%a4%b6%e0%a4%bf%e0%a4%a4-%e0%a4%a1%e0%a4%bf%e0%a4%b8%e0%a5%8d%e0%a4%95-%e0%a4%95\/","title":{"rendered":"\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0948\u0938\u0947 \u0915\u0930\u0947\u0902?"},"content":{"rendered":"<p class=\"sidekick\">\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0948\u0938\u0947 \u0915\u0930\u0947\u0902: \u0938\u0930\u0932 \u0935\u093f\u0927\u093f \u0914\u0930 \u0938\u0942\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u091a\u0930\u0923-\u0926\u0930-\u091a\u0930\u0923 \u092e\u093e\u0930\u094d\u0917\u0926\u0930\u094d\u0936\u0928\u0964 \u092d\u094c\u0924\u093f\u0915\u0940 \u0915\u0947 \u091b\u093e\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940\u0964<\/p>\n<h2>\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0948\u0938\u0947 \u0915\u0930\u0947\u0902?<\/h2>\n<p>\n\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 (Electric Field) \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0930\u0928\u093e \u090f\u0915 \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0938\u092e\u0938\u094d\u092f\u093e \u0939\u0948 \u091c\u094b \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u091a\u0941\u0902\u092c\u0915\u0924\u094d\u0935 (Electromagnetism) \u0915\u0947 \u0905\u0927\u094d\u092f\u092f\u0928 \u092e\u0947\u0902 \u092a\u094d\u0930\u0938\u094d\u0924\u0941\u0924 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u0907\u0938 \u0932\u0947\u0916 \u092e\u0947\u0902, \u0939\u092e \u0907\u0938 \u092a\u094d\u0930\u0915\u094d\u0930\u093f\u092f\u093e \u0915\u094b \u0906\u0938\u093e\u0928 \u0924\u0930\u0940\u0915\u0947 \u0938\u0947 \u0938\u092e\u091d\u0928\u0947 \u0915\u0940 \u0915\u094b\u0936\u093f\u0936 \u0915\u0930\u0947\u0902\u0917\u0947\u0964\n<\/p>\n<h2>\u091a\u0930\u0923 1: \u0921\u093f\u0938\u094d\u0915 \u0915\u093e \u0938\u094d\u0925\u093e\u092a\u0928\u093e<\/h2>\n<p>\n\u092e\u093e\u0928 \u0932\u0940\u091c\u093f\u090f, \u0939\u092e\u093e\u0930\u0947 \u092a\u093e\u0938 \u0915\u0941\u0932 \u0906\u0935\u0947\u0936 \\( Q \\) \u0915\u0947 \u0938\u093e\u0925 \u090f\u0915 \u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0939\u0948 \u091c\u093f\u0938\u0915\u0940 \u0924\u094d\u0930\u093f\u091c\u094d\u092f\u093e \\( R \\) \u0939\u0948\u0964 \u0939\u092e \u0907\u0938 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u0947\u0902\u0926\u094d\u0930 \u092a\u0930 \u090f\u0915 \u0935\u093f\u092d\u0947\u0926\u0928 (coordinate system) \u0915\u093e \u0909\u092a\u092f\u094b\u0917 \u0915\u0930\u0947\u0902\u0917\u0947 \u0914\u0930 \u0939\u092e \u0909\u0938 \u092c\u093f\u0902\u0926\u0941 \u092a\u0930 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u092a\u0924\u093e \u0932\u0917\u093e\u0928\u093e \u091a\u093e\u0939\u0947\u0902\u0917\u0947 \u091c\u094b \u0921\u093f\u0938\u094d\u0915 \u0915\u0940 \u0927\u0941\u0930\u0940 (axis) \u092a\u0930 \u0909\u0938\u0915\u0940 \u0938\u0924\u0939 \u0938\u0947 \\( z \\) \u0926\u0942\u0930\u0940 \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u0939\u0948\u0964\n<\/p>\n<h2>\u091a\u0930\u0923 2: \u091b\u094b\u091f\u0947 \u0924\u0924\u094d\u0935 \u0915\u093e \u0935\u093f\u091a\u093e\u0930<\/h2>\n<p>\n\u0907\u0938 \u0921\u093f\u0938\u094d\u0915 \u0915\u094b \u091b\u094b\u091f\u0947-\u091b\u094b\u091f\u0947 \u0930\u093f\u0902\u0917\u094d\u0938 (rings) \u092e\u0947\u0902 \u0935\u093f\u092d\u093e\u091c\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964 \u090f\u0915 \u091b\u094b\u091f\u0940 \u0930\u093f\u0902\u0917 \u091c\u093f\u0938\u0915\u093e \u0906\u0902\u0924\u0930\u093f\u0915 \u0924\u094d\u0930\u093f\u091c\u094d\u092f\u093e \\( r \\) \u0914\u0930 \u091a\u094c\u0921\u093c\u093e\u0908 \\( dr \\) \u0939\u094b\u0924\u0940 \u0939\u0948, \u0915\u094b \u0926\u0947\u0916\u0924\u0947 \u0939\u0948\u0902\u0964 \u0907\u0938 \u0930\u093f\u0902\u0917 \u092a\u0930 \u0909\u092a\u0938\u094d\u0925\u093f\u0924 \u091b\u094b\u091f\u0947 \u0924\u0924\u094d\u0935 \u0915\u094b \\( dQ \\) \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0928\u093f\u0930\u0942\u092a\u093f\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964\n<\/p>\n<ul>\n<li>\\( dA = 2 \\pi r \\, dr \\) (\u091b\u094b\u091f\u0940 \u0930\u093f\u0902\u0917 \u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932)<\/li>\n<li>\\( \\sigma = \\frac{Q}{\\pi R^2} \\) (\u0938\u0924\u0939\u0940 \u0906\u0935\u0947\u0936 \u0918\u0928\u0924\u094d\u0935), \u092f\u0939\u093e\u0902 \\( \\pi R^2 \\) \u092a\u0942\u0930\u0940 \u0921\u093f\u0938\u094d\u0915 \u0915\u093e \u0915\u094d\u0937\u0947\u0924\u094d\u0930\u092b\u0932 \u0939\u0948\u0964<\/li>\n<li>\\( dQ = \\sigma \\, dA = \\sigma \\cdot 2 \\pi r \\, dr \\)<\/li>\n<\/ul>\n<h2>\u091a\u0930\u0923 3: \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0947 \u0935\u093f\u092d\u093f\u0928\u094d\u0928 \u0918\u091f\u0915\u094b\u0902 \u0915\u093e \u0935\u093f\u091a\u093e\u0930<\/h2>\n<p>\n\u0907\u0938\u0915\u0947 \u092c\u093e\u0926, \u0939\u092e\u0947\u0902 \u092f\u0939 \u0926\u0947\u0916\u0928\u093e \u0939\u094b\u0917\u093e \u0915\u093f \u0930\u0940\u0922\u093c \u0915\u0940 \u0927\u0941\u0930\u0940 \u092a\u0930 \\( z \\) \u0926\u0942\u0930\u0940 \u092a\u0930 \u0938\u094d\u0925\u093f\u0924 \u092c\u093f\u0902\u0926\u0941 \\( P \\) \u092a\u0930 \u0909\u0924\u094d\u092a\u0928\u094d\u0928 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u094d\u092f\u093e \u0939\u094b\u0917\u093e\u0964 \u091b\u094b\u091f\u0947 \u0930\u093f\u0902\u0917 \u0915\u0947 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u091b\u094b\u091f\u0947 \u0924\u0924\u094d\u0935 \u0938\u0947 \u092c\u093f\u0902\u0926\u0941 \\( P \\) \u092a\u0930 \u0926\u093f\u0936\u093e \u0914\u0930 \u092a\u0930\u093f\u092e\u093e\u0923 \u092e\u0947\u0902 \u092a\u094d\u0930\u092d\u093e\u0935 \u092a\u0921\u093c\u0924\u093e \u0939\u0948\u0964\n<\/p>\n<ul>\n<li>\u0930\u093f\u0902\u0917 \u0915\u0947 \u0915\u0947\u0902\u0926\u094d\u0930 \u0938\u0947 \u092c\u093f\u0902\u0926\u0941 \\( P \\) \u0924\u0915 \u0915\u0940 \u0926\u0942\u0930\u0940 \u0939\u0948 \\(\\sqrt{r^2 + z^2}\\)<\/li>\n<li>\u0924\u092c \u0915\u093f\u0938\u0940 \u091b\u094b\u091f\u0947 \u0924\u0924\u094d\u0935 \u0915\u0947 \u0915\u093e\u0930\u0923 \u092c\u093f\u0902\u0926\u0941 \\( P \\) \u092a\u0930 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u092f\u094b\u0917\u0926\u093e\u0928, \\( dE = \\frac{1}{4 \\pi \\epsilon_0} \\frac{dQ}{(r^2 + z^2)} \\) \u0939\u094b\u0917\u093e\u0964<\/li>\n<\/ul>\n<p>\n\u0907\u0938\u0947 z \u0926\u093f\u0936\u093e \u092e\u0947\u0902 \u0918\u091f\u0915\u093f\u0924 (component) \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:\n<\/p>\n<ul>\n<li>\\( dE_z = dE \\cdot \\cos \\theta = \\frac{ \\frac{dQ}{4 \\pi \\epsilon_0 (r^2 + z^2)} \\cdot z}{\\sqrt{r^2 + z^2}} \\) \\rightarrow <math> = \\frac{dQ \\cdot z}{4 \\pi \\epsilon_0 (r^2 + z^2)^{3\/2}} <\/math><\/li>\n<\/ul>\n<h2>\u091a\u0930\u0923 4: \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u0938\u092e\u094d\u092a\u0942\u0930\u094d\u0923 \u0909\u0928\u094d\u200d\u092e\u0942\u0932\u0928<\/h2>\n<p>\n\u0905\u092c, kompleete \u0930\u093f\u0902\u0917 \u092a\u0930 integration \u0915\u0930\u0915\u0947 \u0938\u092e\u094d\u092a\u0942\u0930\u094d\u0923 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948:\n<\/p>\n<ul>\n<li>\\( E_z = \\int_{0}^{R} \\frac{ \\sigma \\cdot z \\cdot 2 \\pi r \\, dr }{ 4 \\pi \\epsilon_0 ( r^2 + z^2 )^{3\/2} } \\)<\/li>\n<\/ul>\n<p>\n\u091c\u0948\u0938\u093e \u0915\u093f \\(\\sigma = \\frac{Q}{\\pi R^2} \\):\n<\/p>\n<ul>\n<li>\\( E_z = \\frac{Qz}{2 \\pi \\epsilon_0 R^2} \\int_{0}^{R} \\frac{r \\, dr}{(r^2 + z^2)^{3\/2}} \\)<\/li>\n<\/ul>\n<h2>\u091a\u0930\u0923 5: \u0938\u092e\u093e\u0927\u093e\u0928<\/h2>\n<p>\n\u0907\u0938 \u0905\u0902\u0924\u093f\u092e \u0907\u0902\u200d\u091f\u0940\u0917\u094d\u0930\u0932 \u0915\u094b \u0939\u0932 \u0915\u0930\u0928\u0947 \u0915\u0947 \u092c\u093e\u0926, \u0939\u092e\u0947\u0902 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u093e \u092a\u0930\u093f\u0923\u093e\u092e \u092e\u093f\u0932\u0947\u0917\u093e:\n<\/p>\n<ul>\n<li>\\( E_z = \\frac{Q}{2 \\pi \\epsilon_0 R^2} \\left(1 &#8211; \\frac{z}{\\sqrt{R^2 + z^2}}\\right) \\)<\/li>\n<\/ul>\n<p>\n\u0907\u0938 \u092a\u094d\u0930\u0915\u093e\u0930, \u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0915\u093f\u0938\u0940 \u092c\u093f\u0902\u0926\u0941 \u092a\u0930 \u0927\u0941\u0930\u0940 \u092a\u0930 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u091c\u094d\u091e\u093e\u0924 \u0915\u093f\u092f\u093e \u091c\u093e \u0938\u0915\u0924\u093e \u0939\u0948\u0964\n<\/p>\n<h2>Summary<\/h2>\n<p><img decoding=\"async\" src=\"https:\/\/www.electricity-magnetism.org\/wp-content\/uploads\/2024\/08\/\u0938\u092e-\u0928_\u0930-\u092a_\u0938_\u0906\u0935-\u0936-\u0924_\u0921-\u0938-\u0915_\u0915_\u0915-\u0930\u0923_\u0935-\u0926-\u092f-\u0924_\u0915-\u0937-\u0924-\u0930_\u0915_\u0917\u0923\u0928_\u0915-\u0938_\u0915\u0930.png\" alt=\"\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0948\u0938\u0947 \u0915\u0930\u0947\u0902?\" style=\"display: block; margin-left: auto; margin-right: auto;\"\/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u0938\u092e\u093e\u0928 \u0930\u0942\u092a \u0938\u0947 \u0906\u0935\u0947\u0936\u093f\u0924 \u0921\u093f\u0938\u094d\u0915 \u0915\u0947 \u0915\u093e\u0930\u0923 \u0935\u093f\u0926\u094d\u092f\u0941\u0924 \u0915\u094d\u0937\u0947\u0924\u094d\u0930 \u0915\u0940 \u0917\u0923\u0928\u093e \u0915\u0948\u0938\u0947 \u0915\u0930\u0947\u0902: \u0938\u0930\u0932 \u0935\u093f\u0927\u093f \u0914\u0930 \u0938\u0942\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u093e\u0925 \u091a\u0930\u0923-\u0926\u0930-\u091a\u0930\u0923 \u092e\u093e\u0930\u094d\u0917\u0926\u0930\u094d\u0936\u0928\u0964 \u092d\u094c\u0924\u093f\u0915\u0940 \u0915\u0947 \u091b\u093e\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f 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