{"id":45433,"date":"2026-04-30T07:47:36","date_gmt":"2026-04-29T23:47:36","guid":{"rendered":"https:\/\/www.qiusir.com\/?p=45433"},"modified":"2026-04-30T22:45:23","modified_gmt":"2026-04-30T14:45:23","slug":"%e5%85%a8%e5%8f%8d%e5%b0%84%e3%80%81%e9%80%8f%e9%95%9c%e3%80%81%e8%99%b9%e5%92%8c%e9%9c%93","status":"publish","type":"post","link":"https:\/\/www.qiusir.com\/?p=45433","title":{"rendered":"\u5168\u53cd\u5c04\u3001\u900f\u955c\u3001\u8679\u548c\u9713"},"content":{"rendered":"

JQX\/\u8fdb\u53d6\u82af \u5e2d\u660e\u7eb3\u7b2c26\u671f\uff082026.03.04\uff09
\n\"\"
\n\u4e00\u3001\u4e00\u4e9b\u5149\u5b66\u7ed3\u8bba\u548c\u516c\u5f0f
\u6298\u5c04\u7387\u7684\u8868\u8fbe\u5f0f\u4e3a $n = \\sin\\theta_i \/ \\sin\\theta_r$
\u5982\u679c\u4e24\u4e2a\u4ecb\u8d28\u5747\u4e3a\u975e\u771f\u7a7a\uff0c\u5219\u6709 $n_1 \\sin\\theta_1 = n_2 \\sin\\theta_2$\uff1b
\u5149\u901f\u4e4b\u6bd4 $v_1\/v_2 = \\frac{c\/n_1}{c\/n_2} = \\frac{n_2}{n_1}$\uff0c
\u6ce2\u957f\u4e4b\u6bd4 $\\lambda_1\/\\lambda_2 = \\frac{v_1\/f}{v_2\/f} = v_1\/v_2 = \\frac{n_2}{n_1}$\u3002
\n\u4e8c\u3001\u5168\u53cd\u5c04
\n1.\u4f8b\u9898
\u5149\u7ea4\u76f4\u5f84\u4e3a $d$\uff0c\u7ed5\u534a\u5f84\u4e3a $r$ \u7684\u5706\u67f1\u8f6c\u5f2f\uff0c\u5982\u56fe\u6240\u793a\uff0c\u82e5\u5149\u7ea4\u4ecb\u8d28\u6298\u5c04\u7387\u4e3a $n$\uff0c\u80fd\u4ee4\u5149\u7ebf\u5728\u5149\u7ea4\u4e2d\u8fde\u7eed\u5168\u53cd\u5c04\u7684\u6700\u5c0f\u5706\u67f1\u534a\u5f84\u4e3a\u591a\u5927\uff08\u8bbe\u5149\u7ea4\u5468\u56f4\u4e3a\u7a7a\u6c14\uff09\uff1f
\n\"\"\u5982\u56fe\uff0c\u6cbf\u7740 AB\u884c\u8fdb\u7684\u5149\uff0c\u5165\u5c04\u89d2\u6700\u5c0f\uff0c\u6700\u4e0d\u6613\u53d1\u751f\u5168\u53cd\u5c04\u3002
\u6545\u5149\u7ebfAB \u5728 $B$ \u5904\u4ea7\u751f\u5168\u53cd\u5c04\u7684\u6761\u4ef6\u4e3a\u6781\u503c\uff1a
$n \\times \\sin \\theta \\geq 1 \\times \\sin 90^\\circ \\implies n \\sin \\theta \\geq 1$
\u6839\u636e\u51e0\u4f55\u5173\u7cfb\uff0c$\\sin \\theta = \\frac{r}{r+d}$ \u4ee3\u5165\u4e0a\u5f0f
\u5219 $n \\times \\frac{r}{r+d} \\geq 1 \\implies r \\geq \\frac{d}{n-1}$
\n2.\u6d77\u5e02\u8703\u697c\u548c\u6c99\u6f20\u8703\u666f
\u9996\u5148\u662f\u6d77\u5e02\u8703\u697c\uff0c\u5176\u6210\u56e0\u6e90\u4e8e\u8fd1\u5730\u9762\u7a7a\u6c14\u5c42\u6e29\u5ea6\u68af\u5ea6\u5bfc\u81f4\u7684\u6298\u5c04\u7387\u8fde\u7eed\u53d8\u5316\uff1a\u6d77\u9762\u6e29\u5ea6\u8f83\u4f4e\u3001\u5bc6\u5ea6\u8f83\u5927\uff0c\u6298\u5c04\u7387\u8f83\u9ad8\uff1b\u800c\u4e0a\u5c42\u7a7a\u6c14\u6e29\u5ea6\u5347\u9ad8\uff0c\u6298\u5c04\u7387\u9010\u6e10\u51cf\u5c0f\u3002\u8fd9\u4e00\u6298\u5c04\u7387\u68af\u5ea6\u4f7f\u5149\u7ebf\u8f68\u8ff9\u5448\u5411\u4e0a\u51f8\u7684\u66f2\u7ebf\uff0c\u5f53\u6765\u81ea\u8fdc\u5904\u7269\u4f53\u7684\u5149\u7ebf\u81ea\u4e0b\u800c\u4e0a\u5c04\u5165\u8be5\u68af\u5ea6\u573a\u65f6\uff0c\u8def\u5f84\u53d1\u751f\u8fde\u7eed\u5f2f\u66f2\uff0c\u6700\u7ec8\u4ee5\u5927\u4e8e\u4e34\u754c\u89d2\u7684\u89d2\u5ea6\u5165\u5c04\u81f3\u4f4e\u6298\u5c04\u7387\u5c42\uff0c\u89e6\u53d1\u5168\u53cd\u5c04\uff0c\u4f7f\u89c2\u5bdf\u8005\u770b\u5230\u4f4d\u4e8e\u4e0a\u65b9\u7684\u865a\u50cf
\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u6c99\u6f20\u8703\u666f\u5219\u5448\u73b0\u5728\u4e0b\u65b9\u7684\u865a\u50cf\uff1a\u6b64\u65f6\u5730\u8868\u6e29\u5ea6\u6781\u9ad8\uff0c\u4e0b\u5c42\u7a7a\u6c14\u6298\u5c04\u7387\u663e\u8457\u4f4e\u4e8e\u4e0a\u5c42\uff0c\u5149\u7ebf\u81ea\u4e0b\u800c\u4e0a\u7a7f\u8fc7\u68af\u5ea6\u4ecb\u8d28\u65f6\u5411\u4e0b\u5f2f\u66f2\uff0c\u5728\u89c2\u5bdf\u8005\u773c\u4e2d\u53cd\u5411\u5ef6\u957f\u7ebf\u6c47\u805a\u4e8e\u5730\u9762\u4e4b\u4e0b\uff0c\u5f62\u6210\u5982\u6c34\u6d3c\u822c\u7684\u865a\u50cf\u3002\u6211\u4eec\u4e58\u5750\u6c7d\u8f66\u65f6\u5e38\u4e8e\u524d\u6321\u98ce\u73bb\u7483\u4e0b\u65b9\u89c2\u5bdf\u5230\u8def\u9762\u201c\u79ef\u6c34\u201d\u5e7b\u5f71\uff0c\u5176\u672c\u8d28\u6b63\u662f\u6c99\u6f20\u8703\u666f\u7684\u5178\u578b\u8868\u73b0\u2014\u2014\u5149\u7ebf\u5728\u8fd1\u5730\u8868\u9ad8\u6e29\u8584\u5c42\u4e2d\u53d1\u751f\u5168\u53cd\u5c04\u6240\u81f4\u3002
\n\u4e09\u3001\u900f\u955c
3.\u900f\u955c\u7684\u6210\u50cf\u516c\u5f0f\u662f $\\frac{1}{f} = \\frac{1}{u} + \\frac{1}{v}$\uff0c\u8fd9\u4e2a\u516c\u5f0f\u4e2d $v$ \u548c $f$ \u53ef\u4ee5\u6709\u6b63\u8d1f\uff0c\u8fd9\u53d6\u51b3\u4e8e\u900f\u955c\u7684\u7c7b\u578b\u4e0e\u5149\u8def\u65b9\u5411\u3002
\u4f8b\u9898\uff1a\u8bd5\u6c42 $u+v$ \u7684\u6700\u5c0f\u503c\u3002
\u6211\u4eec\u53ef\u4ee5\u7528\u6743\u65b9\u548c\u4e0d\u7b49\u5f0f\uff08\u4e00\u79cd\u67ef\u897f\u4e0d\u7b49\u5f0f\u7684\u63a8\u5e7f\u5f62\u5f0f\uff09\u6765\u8bc1\u660e\u3002
\u6743\u65b9\u548c\u4e0d\u7b49\u5f0f\uff08\u5bf9\u4e8e\u6b63\u6570\uff09\u7684\u4e00\u79cd\u5e38\u89c1\u5f62\u5f0f\u662f\uff1a $\\frac{a^2}{x} + \\frac{b^2}{y} \\ge \\frac{(a+b)^2}{x+y}$\uff0c\u5f53\u4e14\u4ec5\u5f53 $\\frac{a}{x} = \\frac{b}{y}$ \u65f6\u53d6\u7b49\u3002
\u5df2\u77e5 $\\frac{1}{u} + \\frac{1}{v} = \\frac{1}{f}$\uff0c\u4e14 $u, v, f > 0$\u3002
\u7531\u6743\u65b9\u548c\u4e0d\u7b49\u5f0f\uff0c\u53d6 $a = 1, b = 1$\uff0c\u5219$\\frac{1^2}{u} + \\frac{1^2}{v} \\ge \\frac{(1+1)^2}{u+v}$
\u5373 $\\frac{1}{u} + \\frac{1}{v} \\ge \\frac{4}{u+v}$
\u4ee3\u5165\u5df2\u77e5\u6761\u4ef6 $\\frac{1}{u} + \\frac{1}{v} = \\frac{1}{f}$\uff0c\u5f97 $\\frac{1}{f} \\ge \\frac{4}{u+v}$\u53ef\u5f97 $u+v \\ge 4f$
\u7b49\u53f7\u6210\u7acb\u5f53\u4e14\u4ec5\u5f53 $\\frac{1}{u} = \\frac{1}{v}$\uff0c\u5373 $u = v$\u3002\u6545 $u = 2f$\uff0c\u4ece\u800c $v = 2f$\u3002
\u56e0\u6b64\uff0c$u+v$ \u7684\u6700\u5c0f\u503c\u4e3a $4f$\uff0c\u5f53\u4e14\u4ec5\u5f53 $u = v = 2f$ \u65f6\u53d6\u5230\u3002<\/p>\n

\u5173\u4e8e\u6743\u65b9\u548c\u4e0d\u7b49\u5f0f\u7684\u5e94\u7528\uff0c\u8fd8\u53ef\u4ee5\u8bc1\u660e\u5f02\u79cd\u7535\u8377\u95f4\u7535\u573a\u7684\u6781\u5c0f\u503c\u70b9\u4f4d\u7f6e\u3002
\u4f8b\u9898\uff1a\u8bbe\u4e24\u4e2a\u70b9\u7535\u8377\u5e26\u7535\u91cf\u5206\u522b\u4e3a $+4Q$ \u548c $-Q$\uff0c\u76f8\u8ddd $d$\uff0c\u6c42\u4e24\u7535\u8377\u95f4\u7535\u573a\u7684\u6781\u5c0f\u503c\u70b9\u4f4d\u7f6e\u3002
\u5728\u5b83\u4eec\u8fde\u7ebf\u4e0a\u53d6\u4e00\u70b9 $P$\uff0c\u5230\u6b63\u7535\u8377\u7684\u8ddd\u79bb\u4e3a $x$\uff0c\u5219\u5230\u8d1f\u7535\u8377\u7684\u8ddd\u79bb\u4e3a $d-x$\uff08$0 < x < d$\uff09\u3002\u5728\u4e24\u70b9\u7535\u8377\u4e4b\u95f4\uff0c\u7535\u573a\u65b9\u5411\u76f8\u540c\uff0c\u5408\u573a\u5f3a\u5927\u5c0f\u4e3a
$E = k\\left( \\frac{4Q}{x^2} + \\frac{Q}{(d-x)^2} \\right) = kQ\\left( \\frac{4}{x^2} + \\frac{1}{(d-x)^2} \\right)$\uff0c
\u5176\u4e2d $k$ \u4e3a\u9759\u7535\u529b\u5e38\u91cf\u3002\u4ee4 $u = x$\uff0c$v = d – x$\uff0c\u5219 $u + v = d$\uff0c\u4e14
$E = kQ\\left( \\frac{4}{u^2} + \\frac{1}{v^2} \\right)$\u3002
\u5229\u7528\u6743\u65b9\u548c\u4e0d\u7b49\u5f0f\uff08$p=2$ \u7684\u60c5\u5f62\uff09\uff1a\u5bf9\u4e8e\u6b63\u6570 $a, b, u, v$\uff0c
\u6709$\\frac{a^3}{u^2} + \\frac{b^3}{v^2} \\ge \\frac{(a+b)^3}{(u+v)^2}$\uff0c\u5f53\u4e14\u4ec5\u5f53 $\\frac{a}{u} = \\frac{b}{v}$\u7b49\u53f7\u6210\u7acb\u3002
\u53d6 $a = \\sqrt[3]{4}$\uff0c$b = 1$\uff0c\u5219$\\frac{4}{u^2} + \\frac{1}{v^2} = \\frac{(\\sqrt[3]{4})^3}{u^2} + \\frac{1^3}{v^2} \\ge \\frac{(\\sqrt[3]{4} + 1)^3}{(u+v)^2} = \\frac{(1+\\sqrt[3]{4})^3}{d^2}$\u3002
\u56e0\u6b64$E \\ge kQ \\cdot \\frac{(1+\\sqrt[3]{4})^3}{d^2}$\u3002
\u7b49\u53f7\u6210\u7acb\u65f6 $\\frac{\\sqrt[3]{4}}{u} = \\frac{1}{v}$\uff0c\u5373 $v = u \/ \\sqrt[3]{4}$\uff0c\u7ed3\u5408 $u+v = d$
\u89e3\u5f97$u = \\frac{\\sqrt[3]{4}}{1+\\sqrt[3]{4}}\\,d,\\quad v = \\frac{1}{1+\\sqrt[3]{4}}\\,d$\u3002
\u6b64\u65f6\u7535\u573a\u5f3a\u5ea6\u53d6\u6700\u5c0f\u503c $E_{\\min} = kQ \\dfrac{(1+\\sqrt[3]{4})^3}{d^2}$\u3002<\/p>\n

4.\u5355\u7403\u9762\u6298\u5c04\u6210\u50cf\u516c\u5f0f
\u5982\u56fe\uff0c\u534a\u5f84\u4e3a $r$ \u7684\u73bb\u7483\uff0c\u4e00\u8fd1\u8f74\u5149\u7ebf\u4ece\u5149\u8f74\u4e0a\u7684 $O$ \u70b9\u51fa\u53d1\uff0c\u7ecf\u7403\u9762\u6298\u5c04\u540e\u4ea4\u5149\u8f74\u4e8e $I$ \u70b9\u3002
\n\"\"
\n\u7531\u6298\u5c04\u5b9a\u5f8b\uff1a$n_1 \\sin i_1 = n_2 \\sin i_2$
\u56e0 $i_1,i_2$ \u5f88\u5c0f\uff0c\u6545 $\\sin i_1 \\approx i_1$\uff0c$\\sin i_2 \\approx i_2$\uff0c\u5f97\uff1a$n_1 i_1 = n_2 i_2 \\tag{1}$
\u7531\u51e0\u4f55\u5173\u7cfb\uff1a$i_1 = \\alpha + \\theta,\\quad i_2 = \\theta – \\beta$
\u53ef\u5f97\uff1a$n_1(\\alpha + \\theta) = n_2(\\theta – \\beta)$
\u6574\u7406\u5f97\uff1a$n_1 \\alpha + n_2 \\beta = (n_2 – n_1)\\theta \\tag{2}$
\u8fd1\u8f74\u6761\u4ef6\u4e0b $\\delta \\ll u,v,r$\uff0c\u6839\u636e\u5c0f\u89d2\u5ea6\u8fd1\u4f3c\uff1a
$\\alpha \\approx \\tan\\alpha = \\frac{h}{u+\\delta} \\approx \\frac{h}{u},\\quad
\\beta \\approx \\tan\\beta = \\frac{h}{v},\\quad
\\theta \\approx \\tan\\theta = \\frac{h}{r}$
\u5c06 $\\alpha,\\beta,\\theta$ \u4ee3\u5165\u5f97\uff1a$n_1 \\cdot \\frac{h}{u} + n_2 \\cdot \\frac{h}{v} = (n_2 – n_1) \\cdot \\frac{h}{r}$
\u7ea6\u53bb $h$\uff0c\u5f97\u5230\u5355\u7403\u9762\u6298\u5c04\u6210\u50cf\u516c\u5f0f\uff1a${\\frac{n_1}{u} + \\frac{n_2}{v} = \\frac{n_2 – n_1}{r}}$
\u5f53\u7269\u65b9\u4e3a\u7a7a\u6c14\uff08$n_1=1$\uff09\u3001\u50cf\u65b9\u4ecb\u8d28\u6298\u5c04\u7387\u4e3a $n$\uff08\u5373 $n_2=n$\uff09\u65f6\uff0c\u516c\u5f0f\u7b80\u5316\u4e3a\uff1a${\\frac{1}{u} + \\frac{n}{v} = \\frac{n – 1}{r}}$
\n5.\u8584\u900f\u955c\u6210\u50cf\u516c\u5f0f
\u8584\u900f\u955c\uff08\u539a\u5ea6\u8fdc\u5c0f\u4e8e\u7269\u8ddd\u3001\u50cf\u8ddd\u53ca\u7403\u9762\u66f2\u7387\u534a\u5f84\uff0c\u4e24\u7403\u9762\u9876\u70b9\u8fd1\u4f3c\u91cd\u5408\u4e8e\u5149\u5fc3 $O$\uff09\uff0c
\u900f\u955c\u6298\u5c04\u7387\u4e3a $n$\uff0c\u4e24\u4fa7\u5747\u4e3a\u7a7a\u6c14\uff08\u6298\u5c04\u7387 $n_0=1$\uff09\uff0c\u5149\u7ebf\u6ee1\u8db3\u8fd1\u8f74\u6761\u4ef6\uff08$\\sin\\theta \\approx \\tan\\theta \\approx \\theta$\uff09\uff0c\u7b26\u53f7\u7ea6\u5b9a\u4e3a\u5b9e\u7269\u3001\u5b9e\u50cf\u53ca\u7403\u5fc3\u5728\u51fa\u5c04\u4fa7\u65f6\u5bf9\u5e94\u7269\u7406\u91cf\u4e3a\u6b63\u3002
\u8bbe\u7269\u8ddd\u4e3a $u$\uff08\u5b9e\u7269 $u>0$\uff09\uff0c\u50cf\u8ddd\u4e3a $v$\uff08\u5b9e\u50cf $v>0$\uff09\uff0c\u5de6\u7403\u9762\u66f2\u7387\u534a\u5f84\u4e3a $R_1$\uff08\u51f8\u9762\u671d\u5411\u7269\u65b9\uff0c\u7403\u5fc3\u5728\u53f3\u4fa7\uff09\uff0c\u53f3\u7403\u9762\u66f2\u7387\u534a\u5f84\u4e3a $R_2$\uff08\u51f9\u9762\u671d\u5411\u50cf\u65b9\uff0c\u7403\u5fc3\u5728\u5de6\u4fa7\uff09\uff0c\u8584\u900f\u955c\u6210\u50cf\u53ef\u89c6\u4e3a\u5149\u7ebf\u7ecf\u5de6\u3001\u53f3\u4e24\u4e2a\u7403\u9762\u5148\u540e\u6298\u5c04\u7684\u53e0\u52a0\u8fc7\u7a0b\uff0c\u5177\u4f53\u63a8\u5bfc\u5982\u4e0b\uff1a
\u9996\u5148\uff0c\u7269\u70b9 $A$ \u53d1\u51fa\u7684\u5149\u7ebf\u7ecf\u5de6\u7403\u9762\uff08\u7a7a\u6c14\u2192\u73bb\u7483\uff09\u53d1\u751f\u7b2c\u4e00\u6b21\u6298\u5c04\uff0c\u7269\u65b9\u4ecb\u8d28\u4e3a\u7a7a\u6c14\uff08$n_1=1$\uff09\uff0c\u50cf\u65b9\u4ecb\u8d28\u4e3a\u73bb\u7483\uff08$n_2=n$\uff09\uff0c
\u6839\u636e\u5355\u7403\u9762\u6298\u5c04\u516c\u5f0f $\\frac{n_1}{u} + \\frac{n_2}{v_1} = \\frac{n_2 – n_1}{R_1}$\uff0c\u4ee3\u5165 $n_1=1$\u3001$n_2=n$\uff0c
\u53ef\u5f97 $\\frac{1}{u} + \\frac{n}{v_1} = \\frac{n-1}{R_1}$\uff0c
\u5176\u4e2d $v_1$ \u4e3a\u7b2c\u4e00\u6b21\u6298\u5c04\u5728\u73bb\u7483\u4e2d\u7684\u50cf\u8ddd\uff0c\u82e5\u65e0\u53f3\u7403\u9762\uff0c\u5149\u7ebf\u5c06\u5728\u73bb\u7483\u4e2d\u4f1a\u805a\u4e8e $v_1$ \u5904\u3002
\u63a5\u7740\u8fdb\u884c\u7b2c\u4e8c\u6b21\u6298\u5c04\uff0c\u5373\u5149\u7ebf\u7ecf\u53f3\u7403\u9762\uff08\u73bb\u7483\u2192\u7a7a\u6c14\uff09\u7684\u6298\u5c04\uff0c\u6b64\u65f6\u7b2c\u4e00\u6b21\u6298\u5c04\u6240\u6210\u7684\u50cf\u6210\u4e3a\u53f3\u7403\u9762\u7684\u7269\uff0c\u56e0\u900f\u955c\u6781\u8584\uff0c\u53f3\u7403\u9762\u7684\u7269\u8ddd\u4e3a $-u_2=-v_1$\uff08\u865a\u7269\uff0c\u7269\u5728\u53f3\u7403\u9762\u50cf\u65b9\u4e00\u4fa7\uff0c\u6545\u7269\u8ddd\u4e3a\u8d1f\uff09\uff0c\u53f3\u7403\u9762\u7269\u65b9\u4ecb\u8d28\u4e3a\u73bb\u7483\uff08$n_1=n$\uff09\uff0c\u50cf\u65b9\u4ecb\u8d28\u4e3a\u7a7a\u6c14\uff08$n_2=1$\uff09\uff0c
\u4ee3\u5165\u5355\u7403\u9762\u6298\u5c04\u516c\u5f0f $\\frac{n_1}{u_2} + \\frac{n_2}{v} = \\frac{n_2 – n_1}{R_2}$\uff0c\u53ef\u5f97 $\\frac{n}{-v_1} + \\frac{1}{v} = \\frac{1 – n}{R_2}$\u3002
\u5c06\u4e24\u6b21\u6298\u5c04\u7684\u516c\u5f0f\u5de6\u53f3\u5206\u522b\u76f8\u52a0\uff0c\u4e2d\u95f4\u9879 $\\frac{n}{v_1}$ \u4e0e $-\\frac{n}{v_1}$ \u76f8\u4e92\u62b5\u6d88\uff0c\u6574\u7406\u540e\u5f97\u5230 $\\frac{1}{u} + \\frac{1}{v} = \\frac{n-1}{R_1} + \\frac{1 – n}{R_2}$\uff0c
\u8fdb\u4e00\u6b65\u5316\u7b80\u4e3a $\\frac{1}{u} + \\frac{1}{v} = (n-1)\\left( \\frac{1}{R_1} – \\frac{1}{R_2} \\right)$\u3002
\u5f53\u7269\u5728\u65e0\u7a77\u8fdc\uff08$u \\to \\infty$\uff0c\u5e73\u884c\u5149\u5165\u5c04\uff09\u65f6\uff0c\u50cf\u6210\u5728\u900f\u955c\u7126\u70b9 $F$ \u5904\uff0c\u6b64\u65f6\u50cf\u8ddd $v=f$\uff08\u7126\u8ddd\uff09\uff0c\u4ee3\u5165\u4e0a\u5f0f\uff0c\u56e0 $\\frac{1}{\\infty}=0$\uff0c\u53ef\u5f97\u7126\u8ddd\u516c\u5f0f $\\frac{1}{f} = (n-1)\\left( \\frac{1}{R_1} – \\frac{1}{R_2} \\right)$\uff0c
\u56e0\u6b64\u8584\u900f\u955c\u7269\u50cf\u516c\u5f0f\u53ef\u7b80\u5316\u4e3a\u66f4\u7b80\u6d01\u7684\u9ad8\u65af\u516c\u5f0f $\\frac{1}{u} + \\frac{1}{v} = \\frac{1}{f}$\u3002
\n\u56db\u3001\u8679\u548c\u9713\u7684\u5f62\u6210
\n\"\"
\n6.\u8679\uff08\u5149\u5728\u6c34\u73e0\u4e2d\u7684\u4e00\u6b21\u53cd\u5c04\u540e\u51fa\u5c04\uff09
\u592a\u9633\u5149\u4e3a\u5e73\u884c\u5149\uff0c\u5165\u5c04\u5230\u7403\u5f62\u6c34\u6ef4\uff08\u6c34\u7684\u6298\u5c04\u7387 $n \\approx 1.331$\uff09\uff0c\u8bbe\u5165\u5c04\u89d2\u4e3a $i$\uff0c\u6298\u5c04\u89d2\u4e3a $r$\uff0c\u6cd5\u7ebf\u4e3a\u6c34\u6ef4\u7403\u5fc3\u4e0e\u5165\u5c04\u70b9\u7684\u8fde\u7ebf\u3002\u5149\u7ebf\u4ece\u7a7a\u6c14\uff08\u6298\u5c04\u7387 $n_1=1$\uff09\u5165\u5c04\u5230\u6c34\u6ef4\uff08\u6298\u5c04\u7387 $n_2=n$\uff09\uff0c\u7531\u6298\u5c04\u5b9a\u5f8b\u53ef\u77e5 $\\frac{\\sin i}{\\sin r} = n$\uff0c\u5373 $\\sin i = n \\sin r$\u3002
\u5149\u7ebf\u5728\u6c34\u6ef4\u5185\u7ecf\u5386\u4e00\u6b21\u6298\u5c04\u3001\u4e00\u6b21\u5185\u53cd\u5c04\u3001\u4e00\u6b21\u51fa\u5c04\u6298\u5c04\uff0c\u7b2c\u4e00\u6b21\u6298\u5c04\u65f6\u5149\u7ebf\u504f\u6298 $i – r$\uff0c\u6c34\u6ef4\u5185\u53cd\u5c04\u65f6\u504f\u6298 $\\pi – 2r$\uff0c\u7b2c\u4e8c\u6b21\u51fa\u5c04\u6298\u5c04\u65f6\u504f\u6298 $i – r$\uff0c
\u603b\u504f\u5411\u89d2\u4e3a\u8fd9\u4e09\u6b21\u504f\u6298\u4e4b\u548c\uff0c\u5373 $\\delta = (i – r) + (\\pi – 2r) + (i – r) = 2i – 4r + \\pi$\u3002
\u6211\u4eec\u89c2\u6d4b\u5230\u7684\u5f69\u8679\u4ef0\u89d2 $\\theta$\uff0c\u662f\u51fa\u5c04\u5149\u7ebf\u4e0e\u5165\u5c04\u592a\u9633\u5149\u7684\u5939\u89d2\uff0c\u4e5f\u5c31\u662f\u603b\u504f\u5411\u89d2\u7684\u8865\u89d2\uff0c\u5316\u7b80\u540e\u53ef\u5f97 $\\theta = 4r – 2i$\u3002
\u592a\u9633\u5149\u5305\u542b\u5404\u79cd\u5165\u5c04\u89d2\u7684\u5e73\u884c\u5149\uff0c\u53ea\u6709\u5f53\u603b\u504f\u5411\u89d2\u53d6\u6781\u5c0f\u503c\u65f6\uff0c\u5927\u91cf\u5149\u7ebf\u4f1a\u6c47\u805a\u5f62\u6210\u9ad8\u4eae\u5ea6\u7684\u5f69\u8679\uff0c\u56e0\u6b64\u6211\u4eec\u53ea\u9700\u627e\u5230\u603b\u504f\u5411\u89d2\u53d6\u6781\u5c0f\u503c\u65f6\u7684\u89d2\u5ea6\u5373\u53ef\u3002
\u5bf9\u603b\u504f\u5411\u89d2\u5173\u4e8e\u5165\u5c04\u89d2 $i$ \u6c42\u5bfc\u5e76\u4ee4\u5bfc\u6570\u4e3a0\uff0c\u53ef\u5f97 $\\frac{d\\delta}{di} = 2 – 4\\frac{dr}{di} = 0$\uff0c\u8fdb\u4e00\u6b65\u5f97\u51fa $\\frac{dr}{di} = \\frac{1}{2}$\u3002
\u5bf9 $\\sin i = n \\sin r$ \u4e24\u8fb9\u5173\u4e8e $i$ \u6c42\u5bfc\uff0c\u5f97\u5230 $\\cos i = n \\cos r \\cdot \\frac{dr}{di}$\uff0c\u5c06 $\\frac{dr}{di} = \\frac{1}{2}$ \u4ee3\u5165\uff0c\u53ef\u63a8\u51fa $\\cos i = \\frac{n}{2} \\cos r$\u3002
\u8054\u7acb $\\sin i = n \\sin r$ \u4e0e $\\cos i = \\frac{n}{2} \\cos r$\uff0c\u5c06\u4e24\u5f0f\u5e73\u65b9\u76f8\u52a0\uff0c\u5229\u7528 $\\sin^2 x + \\cos^2 x = 1$\uff0c\u4ee3\u5165 $\\sin^2 r = 1 – \\cos^2 r$ \u5316\u7b80\u540e\uff0c
\u53ef\u89e3\u51fa $\\cos r = 2\\sqrt{\\frac{n^2 – 1}{3n^2}}$\u3002
\u5c06\u6c34\u7684\u6298\u5c04\u7387 $n \\approx 1.331$ \u4ee3\u5165\u4e0a\u5f0f\uff0c\u8ba1\u7b97\u5f97\u51fa $\\cos r \\approx 0.7648$\uff0c\u8fdb\u800c\u6c42\u5f97 $r \\approx 40.1^\\circ$\uff1b\u518d\u6839\u636e $\\sin i = n \\sin r$\uff0c
\u53ef\u7b97\u51fa $\\sin i \\approx 0.8573$\uff0c\u5373 $i \\approx 58.9^\\circ$\u3002
\u6700\u540e\u5c06 $i$ \u548c $r$ \u4ee3\u5165\u4ef0\u89d2\u516c\u5f0f $\\theta = 4r – 2i$\uff0c\u8ba1\u7b97\u53ef\u5f97 $\\theta \\approx 42.6^\\circ$\uff0c\u7ea6\u4e3a42\u00b0\u3002
\u7531\u6b64\u53ef\u89c1\uff0c\u5f69\u8679\u4ef0\u89d2\u4ec5\u7531\u6c34\u7684\u6298\u5c04\u7387\u51b3\u5b9a\uff0c\u662f\u56fa\u5b9a\u503c\u3002
\u592a\u9633\u5149\u4f5c\u4e3a\u590d\u8272\u5149\uff0c\u5728\u6c34\u6ef4\u4e2d\u4f20\u64ad\u65f6\u53d1\u751f\u8272\u6563\u73b0\u8c61\uff0c\u4e0d\u540c\u8272\u5149\u7684\u6298\u5c04\u7387 $n$ \u4e0d\u540c\uff0c\u5bfc\u81f4\u6700\u7ec8\u51fa\u5c04\u7684\u4ef0\u89d2 $\\theta$ \u4e0d\u540c\uff0c\u5f62\u6210\u56fa\u5b9a\u7684\u989c\u8272\u987a\u5e8f\u3002
\u8ba1\u7b97\u5f97\u4ef0\u89d2\u7ea6\u4e3a $42.6^\\circ$\uff1b\u7d2b\u5149\u6ce2\u957f\u6700\u77ed\uff0c\u6298\u5c04\u7387\u6700\u5927\uff08\u7ea6 $n_{\\text{\u7d2b}} \\approx 1.342$\uff09\uff0c\u8ba1\u7b97\u5f97\u4ef0\u89d2\u7ea6\u4e3a $40.7^\\circ$\u3002
\u7531\u4e8e\u4ef0\u89d2\u8d8a\u5927\uff0c\u5bf9\u5e94\u7684\u8272\u5149\u5728\u5f69\u8679\u4e0a\u7684\u4f4d\u7f6e\u8d8a\u9760\u5916\uff0c\u56e0\u6b64\u4e3b\u8679\u5448\u73b0\u5916\u7ea2\u5185\u7d2b\u7684\u989c\u8272\u987a\u5e8f\uff0c\u8fd9\u662f\u7531\u4e0d\u540c\u8272\u5149\u7684\u6298\u5c04\u7387\u5dee\u5f02\u5bfc\u81f4\u7684\u4ef0\u89d2\u5206\u5316\u51b3\u5b9a\u7684\u3002<\/p>\n

7.\u9713\uff08\u526f\u8679\uff0c\u5149\u5728\u6c34\u73e0\u4e2d\u7684\u4e24\u6b21\u53cd\u5c04\u540e\u51fa\u5c04\uff09
\u526f\u8679\u7531\u5149\u7ebf\u5728\u6c34\u6ef4\u5185\u7ecf\u5386\u4e24\u6b21\u5185\u53cd\u5c04\u5f62\u6210\uff0c\u5176\u5149\u8def\u4e3a\uff1a\u5165\u5c04\u6298\u5c04 $\\to$ \u7b2c\u4e00\u6b21\u5185\u53cd\u5c04 $\\to$ \u7b2c\u4e8c\u6b21\u5185\u53cd\u5c04 $\\to$ \u51fa\u5c04\u6298\u5c04\u3002
\u8bbe\u5165\u5c04\u89d2\u4e3a $i$\uff0c\u6298\u5c04\u89d2\u4e3a $r$\uff0c\u6bcf\u6b21\u6298\u5c04\u7684\u504f\u6298\u89d2\u4e3a $i-r$\uff0c\u6bcf\u6b21\u5185\u53cd\u5c04\u7684\u504f\u6298\u89d2\u4e3a $\\pi-2r$\uff0c
\u5219\u603b\u504f\u5411\u89d2 $\\delta$ \u4e3a\u5404\u9636\u6bb5\u504f\u6298\u89d2\u4e4b\u548c\uff0c\u5373 $\\delta = (i-r) + 2(\\pi-2r) + (i-r) = 2\\pi + 2i – 6r$\u3002
\u89c2\u6d4b\u5230\u7684\u526f\u8679\u4ef0\u89d2 $\\theta$ \u4e3a\u5165\u5c04\u5149\u7ebf\u4e0e\u51fa\u5c04\u5149\u7ebf\u7684\u5939\u89d2\uff0c\u6ee1\u8db3 $\\theta = \\delta – \\pi = \\pi + 2i – 6r$\u3002
\u4e0e\u4e3b\u8679\u63a8\u5bfc\u903b\u8f91\u4e00\u81f4\uff0c\u526f\u8679\u7684\u53ef\u89c1\u6027\u6e90\u4e8e\u504f\u5411\u89d2\u53d6\u6781\u503c\uff0c\u5bf9 $\\delta$ \u5173\u4e8e $i$ \u6c42\u5bfc\u5e76\u4ee4\u5bfc\u6570\u4e3a0\uff0c\u53ef\u5f97 $\\frac{d\\delta}{di} = 2 – 6\\frac{dr}{di} = 0$\uff0c
\u5373 $\\frac{dr}{di} = \\frac{1}{3}$\u3002
\u5bf9\u65af\u6d85\u5c14\u5b9a\u5f8b $\\sin i = n \\sin r$ \u4e24\u8fb9\u6c42\u5bfc\uff0c\u5f97 $\\cos i = n \\cos r \\cdot \\frac{dr}{di}$\uff0c\u4ee3\u5165\u6781\u503c\u6761\u4ef6\u5f97 $\\cos i = \\frac{n}{3} \\cos r$\u3002
\u8054\u7acb $\\sin i = n \\sin r$ \u4e0e $\\cos i = \\frac{n}{3} \\cos r$\uff0c\u5e73\u65b9\u76f8\u52a0\u540e\u5316\u7b80\u5f97 $\\cos r = \\sqrt{\\frac{9(n^2-1)}{8n^2}} = \\frac{3}{2\\sqrt{2}}\\frac{\\sqrt{n^2-1}}{n}$\u3002
\u4ee3\u5165\u6c34\u7684\u6298\u5c04\u7387 $n \\approx 1.33$ \u8ba1\u7b97\uff0c\u53ef\u5f97 $r \\approx 45.6^\\circ$\uff0c$i \\approx 71.9^\\circ$\uff0c\u518d\u4ee3\u5165\u4ef0\u89d2\u516c\u5f0f $\\theta = \\pi + 2i – 6r$\uff0c
\u8ba1\u7b97\u5f97\u526f\u8679\u4ef0\u89d2\u7ea6\u4e3a $50.1^\\circ \\sim 53.4^\\circ$\uff08\u4e0d\u540c\u8272\u5149\u7565\u6709\u5dee\u5f02\uff09\uff0c\u8fd9\u5c31\u662f\u526f\u8679\u7684\u56fa\u5b9a\u89c2\u6d4b\u89d2\u5ea6\u3002<\/p>\n

\u4ece\u4ef0\u89d2\u516c\u5f0f\u7684\u672c\u8d28\u6765\u770b\uff0c\u4e3b\u8679\u4ef0\u89d2 $\\theta_1 = 4r – 2i$\uff0c\u526f\u8679\u4ef0\u89d2 $\\theta_2 = 6r – 2i – \\pi$\u3002
\u7ed3\u5408\u6298\u5c04\u7387\u4e0e\u89d2\u5ea6\u7684\u5173\u7cfb\u53ef\u77e5\uff0c\u4e3b\u8679\u4e2d\u6298\u5c04\u7387\u8d8a\u5c0f\u7684\u7ea2\u5149\uff08$n$ \u5c0f\uff09\uff0c\u4ef0\u89d2\u8d8a\u5927\uff08\u4f4d\u4e8e\u5916\u4fa7\uff09\uff1b\u800c\u526f\u8679\u7684\u4ef0\u89d2\u516c\u5f0f\u7ed3\u6784\u4e0e\u4e3b\u8679\u4e92\u8865\uff0c\u6298\u5c04\u7387\u8d8a\u5c0f\u7684\u7ea2\u5149\uff0c\u8ba1\u7b97\u51fa\u7684\u4ef0\u89d2\u53cd\u800c\u8d8a\u5c0f\uff08\u4f4d\u4e8e\u5185\u4fa7\uff09\uff0c\u56e0\u6b64\u526f\u8679\u5448\u73b0\u5916\u7d2b\u5185\u7ea2\u7684\u8272\u5e8f\uff0c\u4e0e\u4e3b\u8679\u5b8c\u5168\u76f8\u53cd\u3002
\u526f\u8679\u56e0\u7ecf\u5386\u4e24\u6b21\u5185\u53cd\u5c04\uff0c\u5149\u80fd\u635f\u5931\u66f4\u591a\uff0c\u4eae\u5ea6\u663e\u8457\u4f4e\u4e8e\u4e3b\u8679\uff0c\u4e14\u5176\u4ef0\u89d2\u8303\u56f4\u4e0e\u4e3b\u8679\u5b58\u5728\u91cd\u53e0\u533a\u57df\uff0c\u5171\u540c\u6784\u6210\u4e86\u53cc\u5f69\u8679\u666f\u89c2\u3002
\u4ece\u516c\u5f0f\u5c42\u9762\u4e5f\u53ef\u8bc1\u660e\uff0c\u53ea\u6709\u5f53\u6c34\u6ef4\u6298\u5c04\u7387\u6ee1\u8db3\u4e00\u5b9a\u6761\u4ef6\u65f6\uff0c\u4e3b\u3001\u526f\u8679\u7684\u6781\u503c\u89d2\u5ea6\u624d\u4f1a\u540c\u65f6\u5b58\u5728\uff0c\u8fd9\u5c31\u662f\u6211\u4eec\u80fd\u89c2\u6d4b\u5230\u8679\u4e0e\u9713\u76f8\u4f34\u51fa\u73b0\u7684\u7269\u7406\u4f9d\u636e\u3002<\/p>\n

\u8fd9\u91cc\u6709\u4e00\u4e2a\u6a21\u62df\u52a8\u753b\uff0c\u6a21\u62df\u4e86\u4e0d\u540c\u8272\u5149\u5728\u4e0d\u540c\u5165\u5c04\u9ad8\u5ea6\u7684\u51fa\u5c04\u5149\u8def\uff0c\u53ef\u4ee5\u53c2\u8003\uff1ahttps:\/\/assets.qiusir.com\/rainbow4.html<\/a>
\n

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JQX|Jin<\/center>
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\n\u5047\u8bbe\u5730\u7403\u6709\u4e00\u4e2a\u8d2f\u7a7f\u7684\u6d1e\uff0c\u4ece\u6d1e\u7684\u4e0a\u7a7a\u4e0b\u843d\u4e00\u4e2a\u5c0f\u7403\u5b8c\u6210\u4e00\u6b21\u5b8c\u6574\u5468\u671f\u9700\u8981\u591a\u4e45\u65f6\u95f4\uff1f\u4ece\u7b80\u8c10\u632f\u52a8\u5230\u5f00\u666e\u52d2\u65b9\u7a0b\u6c42\u5c0f\u7403\u7684\u8fd0\u52a8\u5468\u671f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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