Physics-
General
Easy
Question
The diagram shows three concentric conducting spherical shells having radii T, 2R and 3R. The initial potential of each shell is as mentioned in the figure. If the inner most shell is earth then the charge present on its outer surface would be equal to
![](data:image/png;base64,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)
- zero
The correct answer is: zero
Related Questions to study
physics-
The variation of gravitational field E and potential
as a function of x from the center of a uniform spherical mass distribution of radius R is correctly represented, from the following plots, by
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARkAAACNCAYAAACDkA72AAAgAElEQVR4nO3deVTN+f8H8GebUoSakZQtSUhGRkqkLGWYGUvWScRBZDRkzBzGLKjGOmUdnEkbDbJkb6IsFZEWRanUaGEm6kqb6nZ7//7w6341Wm51P/fe6vU4p3O6n8/7836/LvXq83nf9yLHGGMghBCOyEs7AEJI20ZJhhDCKUoyhBBOUZIhhHCKkgwhhFOUZAghnKIkQwjhFCUZ0i7cunULPB5P2mG0S5RkSLvw+PFjlJSU1Hs+LS0N33zzjQQjaj8oyZA2TyAQ4Pnz58jKyqrzfHV1NZYsWYK9e/fi+vXrEo6u7Wt2kikrK4O9vT0EAoE44yFErCorK2Fvb4+qqipcuXIFBw4c+KDM77//jqioKADA0qVL8fbtW0mH2aY1O8n88MMPCAwMhKenpzjjIUSsYmNjIS8vjz59+mDp0qXYvHnzB2Xu3buHHj16wMDAAFlZWcjLy5NCpG1Xs5LMvXv3sHfvXgDAjz/+iIyMDLEGRYi4VFdXC79XUlKCiorKB2WUlJTg4eGBgQMHYvLkyZIMr11odpJRUlKCuro6KisrER8fL+64CBGLTz/9FKWlpSgrK0NcXBzs7e0/KLN+/XoYGxtDTk4Ofn5+UFdXl0KkbVezH5d27tyJTp06Yf369eKMhxCxUlZWxtGjR/HkyRMUFxdjy5YtH5QxNDQUft+9e3doaGhIMsQ2r1lJZuLEiVi9ejUAwM3NDcbGxmINihBx0tTUhImJCcaNGwclJSVph9PuNCvJDB48WPi9oqIiDAwMxBYQIVwYO3YsunXrJu0w2iXF+k5ERERg9uzZmDVrFkaPHg3GGOLj46Guro6ffvpJkjES0mJDhw6VdgjtVr1JZuzYsejZsydGjhyJr776CgAwZ84cnDhxQmLBEUJavwYfl+Tk5Gq9Li8vh4ODA6cBEULalkb7ZAoKCpCTk4P09HT4+flJIiZCSBtS7+NSjZcvXyIjIwNFRUX4559/JBETIaQNaTTJDBo0CFZWVgCA0aNHcx4QIaRtadJH2B999BEePXqEoqIiruIhhLQxDSaZ0tJSlJaWAng3ByQzMxPe3t407JoQIrJ6H5fu3r2LDRs2AAACAgLAGANjDEuWLJFYcFxLTk7GgAEDaBQoIRyqN8mYm5vD3NxckrFIVHV1NT7//HNERERAR0dH2uEQ0ma125XxwsPDMWrUKEowhHCs3SaZAwcOYM2aNdIOg5BGMcZw5coV9O7dGwsXLoSfnx/mz58PHx8fMMakHV6jGv0Iuy0KCwtDdXU1Ro0aJe1QCGmUnJwcpkyZgg4dOmDmzJmYPn06rKys0L9/f6ioqGD+/PnSDrFB7S7JVFVVYd26dTQHi7Q68vL/e/DQ1dVFhw4dkJaWJsWIRNPuHpe2bt2KsWPHwtDQENeuXUN6erq0QyKkyU6cOIEuXbrI/F0M0M7uZKKjo3H69GnExMQAAIKDg2FjY4MBAwZIOTJCRHPz5k0EBgaisrISmZmZda5ZLGvaTZIpKirC4sWLERAQAFVVVQDAmjVraCEj0qpYWVnBwMAAZmZmuH//PsaNGyftkBrVLh6XGGNwcHDA0qVL8emnnwqPBwYGIiEhQYqREdJ0gwcPxs6dOzF37txWMWlZKncyz58/x4EDB5Cfny+cfFlQUIDHjx/j0KFDYm9vy5YtUFNTw7p162odLy4uRmVlpdjbI0TcGGN48+aNcN7g8uXLERISgrlz5+LSpUvNnurz9u1b/Pjjj0hNTYWGhgbc3NzQq1cvAMDDhw+FfZZqamqYMGECBAIBQkNDwefzoaurCzMzs0bbkEqS0dHRgZaWFiIiInDkyBHh8bo23mqpEydO4Pz584iMjPzg3Lp169CpUyext0lkU1ZWFrKzs+s8l5qairy8PEREREg4qsYxxpCYmIgFCxYgMzMTFy5cQLdu3bB06VKEh4fDy8sLo0aNEnYDNNWkSZOQlJQEf39/BAcHw9PTE4sXL8aQIUNw48YNrF27FmfOnEGHDh0AAK9evUJaWhqmT58uUv1S65OpCfh93333nVjbCA4OhpubG8LCwur8D3Bzc4ONjQ2mTZsm1naJbLp37x7OnTtX5zkej4fExETY2NhAXV0d3bp1Q7du3aCpqYnOnTtLONKGpaam1nr94sULpKSkIDk5udYi/01RVVUFFRUVFBUV4ciRI5gyZQp69OiBNWvWIDs7G9u3b8e0adNQUlKC+Ph47NixA4qKIqYP1gI9e/Zs9rUHDx5kY8aMEb6+f/8+y8/Pb0k4tVy5coUZGhqy3Nzcesv4+/uz2NhYsbVJWq8HDx4wfX19NmLECBYcHMx8fX2Zi4sLMzU1ZTo6Omz69Ols586d7P79+6y6ulra4dZJR0enyddUVVWxrVu3Mnl5edahQwe2bds2xufza5UpKytj+vr6bMeOHWzlypXs5cuXTWpDqklmwIABzN/fn/n4+LAxY8aILcn4+voyQ0ND9uzZswbLXb16laWmpoqlTdK6PXjwgH355ZcsNDSU6evrMy8vL2EyKS0tZTdu3GBubm7M2tqa6erqssWLF7MzZ86w4uJiKUf+PxEREU2+pry8nGVnZ7Ps7OwGf//Cw8OZnJwcO3r0aJPbkJk7maysLPbmzZtaZbKzs1lQUFCT6v3555+ZmZmZSNnW2dmZBQcHN6l+0jbVJBnGGOPxeGz+/PlswoQJLDs7+4OyBQUFzN/fn9nZ2TFNTU02Y8YMdvLkSVZaWirpsGuxsLDgrO5bt26x4cOHM0NDQ1ZWVtaka2XmI+zevXtDXV0dWVlZAN4txbBhwwbcuXNHpOtrdlJ4+PAhwsPD8fHHHzd6jZOTE0xNTVsUN2l7unXrhsDAQCxbtgxWVlY4duxYrfMaGhpwcHDA6dOnkZ2djXnz5uHkyZPo06cPFixYgNu3b0sl7venHYjTixcvEBYWhtu3b0MgEAjXmRI5Lk6iEkFVVdUHx65fv46cnBwAwKlTpzB58mSR6srOzsaYMWPQu3dvnDlzBh07dhTpupCQkFYx94NIx9y5cxEREYFjx45h9uzZKCgo+KCMqqoq5syZgzNnziAzMxPjx4/Ht99+iyFDhmDPnj0SXarWw8ND7HW+ffsWu3fvxsaNG9GpUyf8+eefOHToEMLDw0WuQypJJjc3F8XFxRgwYAB8fHwQEBCA/fv34+LFi7CwsEBaWho0NTXRs2fPRusKCwuDtbU1fvjhB7i7uzcpm2dlZaGwsLAlb4W0cT179kRISAisra0xatQoXL16td6ynTt3xpIlS3D//n0cO3ZM+GmPh4cHSkpKOI913rx5Yq3v1q1bGDduXK3fqe7du0NTUxP29va4du2aaBW15DmtJX0yDVm+fDnz9fVl3333HbOxsWGPHj36oEx5eTlzdXVlw4YNY8nJyc1q5/Xr101+viRt0/t9MvVJTU1lo0aNYk5OTqykpESkel++fMl+/PFHNnDgQLZz505Of962b98u1vqqq6trfdV3rDEy0yfzvkmTJkFZWRlKSkpQUFCAgoJCrfMJCQkwNTWFoqIi7t+/D3V1dWzbtg2bNm1CYGAgAgMDsXfvXvz8888NtvPDDz8gNDSUy7dC2hADAwNERUVBR0cHI0eOxN27dxu95uOPP8aWLVsQGxsLxhhGjhyJw4cP19ld0FKvX78Wa31ycnK1vuo71qiWZDqu7mRqhIWFsbVr1wpfV1RUCP8q3L59u1bZ3bt31/q0ijHG1q9f32D9hw8fZtHR0eILmLRaotzJvHnzhv3111/C8kZGRmzjxo2ssrJS5Hby8/OZi4sLGzRoELt8+XKLYv6v5oyTkQSZvJOpYWlpiV9//RUAEBMTg5EjR6KoqAhxcXEYO3ZsrbJ1dfZ+//33DdZvaGiI7t27iy9g0qaVl5fj0aNHAIARI0YgJiYGpaWlMDMzw+PHj0WqQ1NTE3v27MG5c+fg4eEBBwcH8Hg8ka51d3dv8HxD/UXSJNNJRlFREUVFRfj666/h6OiIgwcPwsvLS6Q5GikpKY12Ap88eRKJiYniCpe0cRkZGXj27Bn4fD4AQEVFBV5eXtixYwdmzJiB3377TeQ1dwcOHIjbt29j5MiRGDlyJC5dutToNcXFxQ2eX7BggUhtS5rMJpm3b9/Cw8MDlpaWGDJkCBISEmBhYdHgNXl5eQgICICvry+WLFmC6urqBss7ODhg+PDh4gybtFHu7u54+PAh5OXlMXv2bOHs/YqKClhZWSEmJgbx8fGYMGFCvZMw/0teXh4uLi4IDQ2Fm5sbvvnmG1RUVHxQjjGGixcvIiUlBTdv3qy3vuZOkOSazCWZ8vJy7Nu3D+bm5lBSUkJcXBxWrlwp0gZsWlpacHBwgKOjI3x8fGp1GF+5cgUHDhxAWVmZ8FhMTAxyc3M5eR+kbTl69CimTZuG3r17Q0VFBTdu3MCuXbswf/58DBo0CGlpaQgICICzszOsra3h5+cnct39+/dHREQEOnbsiNGjRyMzM7PW+T179uCff/7BoEGD4O/vjxs3bgjPvf+YtmfPnpa/UQ7ITJIpLS3F7t27YWRkhLy8PNy+fRvr168XeWDdfxkaGqJr1654+fIl4uLioKKiAh6PB1dXV2GZJ0+e4NWrV+J6C6QNe/8xqHPnzigvL4eVlRXOnj2L48ePC5cpmTVrFiIjI3Hq1CnY2dkhPz9fpPqVlJSwbds2eHh4wNbWFtHR0cJzkZGRGDFiBABg/PjxCA4OBgCcPn0aU6ZMEZabOXNmi98nF6SeZJ4+fQpXV1fo6enh6dOniIiIgJubW5MX4anr0Sg+Ph5JSUkYPnw4xo8fDxcXF+GCPACE/6GENGbevHmIjo5GSUkJCgsLMXXqVOEqi/r6+ujdu7ewrLa2Ni5fvgwbGxuYmZnh8uXLIrdja2uLc+fOYdGiRTh9+jSAdwPgagaNVlZWCn+GZ82aBU1NTeG1y5cvb/H75IJU1pPh8/m4dOkSDh8+jMePH2PZsmV4+PAhevTo0az6cnNzkZ2dDXV1dfj4+EBRURFv3rxBdHQ0/Pz8ICcnh8TERPzyyy+1FsbauHEjrSdDROLu7o4zZ84gNzcX+/fvr7WWysWLF+tcC8nJyQkTJkzAokWLcP78efz2228iLZJmZGSEmzdvYurUqSgtLcW6devg5eUFHo+HvLw87Nu3T1j2/bEqonY6S1xLPv9u6jiZBw8esNWrVzMtLS1ma2vLgoKCPli7gisCgYCdPXuWGRoaCo95enqyyMhIibRPZJso42Ty8vLY7t27ax178uQJS09Pb/C6qqoq5u7uzgwNDZv085afn89MTEyYj48Pq6ysZGvWrPmgjImJifB7WR0nw/mdTFJSEk6dOoWgoCAoKChg0aJFiIuLE2lekjjJy8tjxowZCAoKEh6zsLCAtra2ROMgrVfnzp0xadIk4eubN2+ioqICffv2xalTpzBp0qQ6d79QUFDAxo0b8dlnn8HR0RFTpkzB5s2b61wd8n2ampoIDQ2FjY0NFBUVoays3GD5CxcuNO+NcUzsSebvv//GnTt3cOfOHYSHh6NTp06YPn06zp492+ylAVvq2LFj6NGjB3g8HmbNmiU87uvrCxsbG+jq6japvrCwMLi5uUEgEGDOnDlwdnaGvLw8SktLcfnyZZSXlwMARo8eDX19fSQmJiIhIQEKCgr44osvmr3oM5Gujh07YujQoQDeLXzv6+sr/Mi5V69emDNnToPXDx8+HPfu3cPGjRthZmYGf39/GBkZNXiNpqYmQkJCYGlpiZ07d9Y6V11djYqKCjDGICcnBycnJ+GeYrJEjrH6H+SeP3+O+Pj4Os89e/YMrq6u0NfXR0lJCUpKSlBaWgolJSWoq6tDXV0dXbp0aTT7SkJFRQUqKiqgqqpa61m6qqoKixcvbtbdTFJSEnbt2gUejwcrKyscPXoU/fr1w/PnzzFjxgyUl5fj4cOHkJOTA2MM1tbW8Pb2Rv/+/cX51oiYxMbGYsuWLTh//rxE2rtx4wacnJywfPlyuLq6NjpwNC0tDVOnTsWFCxcwaNAgAO8+HU1LS0P//v0xZMgQjBgxArGxsZIIv0kavJPJzc2t9x/91atX4PP5SE1NhbKyMpSVlaGurg45OTlUVVWBx+OJPFxaWqqrq3H58mWRFrj6r4qKCqipqYHH4yE+Ph737t1Dv379oKOjg+DgYBgbGyM4OBgzZszAoUOHsGnTJkowRMja2hoxMTFwcXERjqvp27dvveUNDAzg4+MDOzs73L17F126dIGhoSEMDQ2FZf744w9JhN50LenQ0dbWZkFBQczMzIwZGxszX19fVlFRIY6+IomoWX6zpKSEPXv2TPiVl5fXYIf03bt3WZ8+fRgANnnyZJaTk/NBmYCAAKalpcWCgoLY3r17uXwbRAxE6fjlypkzZ5ienp5I6+d6eXkxOzu7Os/Jasev2D5dioqKYjNmzGC6urrMw8OD8Xi8FgfHtfz8fFZaWsoqKyvZwYMHGQB24sQJdvr0aWZkZMS8vb3rvK64uJi9efOGvXnzpt41Naqrq5mtrS3r06cPEwgEXL4NIgbSTDKMMfbvv/+yqVOnsmnTpjW6PvWMGTPq/MO1adMmrsJrEbF/hJ2ens5WrVrFunfvzr7++mv29OnTljTBqdWrV7Pz588zxhhLTk5m79/YnT17lgFgmZmZzaqbz+czZ2dnpqamxg4dOiSWeAl3pJ1kahw5coTp6ekJfy7r8vr1a2ZgYPDB75a7uzvX4TWL2Ef86uvrY//+/UhJSYG2tjbGjRsHOzs7kRcEb0zNNpmffPIJ3NzccPz4cfj7+8PZ2RknTpxoUl3a2tr1ftLD/r8/vGbGbVPt2rULrq6uOHjwIFxdXfH06dNm1UPal2XLluHatWvYsWMHlixZUufM665du8LT0xPffPNNreMHDx6UVJhN05IMJcpgvIqKCubj48OGDh3KzMzMWFBQEKuqqmpJs4wxxrS0tFhISIjwdXFxMTtw4ECT6khKSmL//vsvY+x/dzICgYClp6czKysr9t133zUrtqCgIHbp0iXG2LvHpq+++opZWFhIbOAhaTpZuZOpIRAI2LZt25iBgQG7detWnWXmzZvHLly4IHwdFRUlqfCaRKIjfv/66y9mY2PD9PT02J49e1q0MVbPnj1rJZnmdDi/v+9STZI5efIk09LSYr6+vk2ur6CggG3ZsoX16tVLOAq0oqKCubu7MxUVFbZq1Sr2zz//NLlewj1ZSzI1EhIS2LBhw9j69etZeXl5rXN5eXlsxIgRwj9e/10ZUlZIZfnNxMRE5ujoyLS0tNj333/f4FayDbVdk2TKy8vZiRMnmlzH5cuXWUpKCmOsdp9MSEgIU1dXp90l2xFZTTKMvfv5/vbbb9mwYcNYQkJCrXNbtmxhv//+O2OMMSMjI2mE1yipzMIeOnQofHx8EB8fDwUFBYwYMQILFy7Ew4cPm1TP9evXERAQgH379iEhIaHJcRQWFgpH577P1tYWS5cuhZ2dXa31ZwiRBmVlZezcuRN79+7F3LlzsW3bNuGqA2vXrsXhw4fx9u1bnDx5UsqR1k2qSz1oa2vD3d0dGRkZMDU1hZ2dHSZOnIirV6+KNKN04sSJcHBwwLfffosVK1YIj588eRKff/45fvrpJwgEgnqvj4qKEu5YmZqaCgDCDloPDw8oKSnB3t4eL1++bMnbJEQsLC0tERMTg7S0NFhaWiIzMxOdOnWCo6Mj9u3bBxsbG2mHWLeW3AaJe7cCgUDAzpw5w0aPHs0GDx7M/vjjjw+eQ99v+/0+GcbezYj9+++/mZ+fHystLWWOjo7My8ur3vZycnJYYWEhKywsZAkJCcKvt2/fMsbejaNJSEhgSUlJ4nuTRCbJ8uNSXYKDg5menh47cuQIKykpYQMHDqxzlrYskPqiVe+Tl5fHzJkzERUVBW9vb4SEhEBPTw9bt279YIWxmolhNYqKinD8+HHo6Ohg4cKFUFVVhaOjY4PbhO7btw937txBly5dMGzYMOGXiooKgHeT04YNG9boJDZCJG3atGm4e/cuLl26hHnz5sHa2hoZGRnSDqtOUlm0ShRmZmYICgrC33//DS8vLwwePBh2dnZwcXHBq1evsGLFCjx+/Bj5+fmoqqrCixcv8Pnnn9daC/j+/ftYtGhRvW0oKyvXmjBJ2jfGGMLCwqQdRpO4uLggJCQEx48fF3mpT0lrcBZ2Y3R0dPD8+XNxxlOvwsJCHD58GPv374eJiQnWrVsHS0vLestHRUVBTk4Oo0ePrrdMZmYmunbtCg0NDS5CJq1IbGwsfvnllzrXg2kNUlJSoKam1uBuBlLTkmctrneQrEtlZSXz9/dnw4YNY59++in7888/PxjklpmZyWJjYxlj74Zg1+f9cTKkfWttfTLvy87OZv369ZPZcTIy1ScjCiUlJTg4OCAhIQHbtm2Dv78/9PX1sXv3bhQVFSEjIwOLFi3Cxo0bMXnyZHh6etZbl7W1NfT09CQYPWmN7t27h/T0dGmHUa8VK1Zg69at+Pfff6UdSp1adYfEhAkTMGHCBCQnJ2P37t3Q19eHg4MDjh07Vmv1+PooKCg0ulgQIS9evBB9c3kJO3LkCBQUFGBvbw9TU1Nph1OnNvEbNnjwYHh7eyMpKQmqqqowNTXF/Pnz8eDBgwavu379Ok1cJA3i8/lITk5GfHx8ozuSSlpaWhq2b98Ob29vAO/uzGVRm0gyNbS0tLB161ZkZmbC0tISX331FcaNG4eLFy/WObhvzZo1jW59S9qvqqoqzJ07F506dcKbN2+wevVq4bm8vLwGB3pyjc/nY8GCBdi/f79wZcfZs2dLLZ6GtKkkU0NVVRUrV67EkydP4Orqip07d2LQoEHC4dc1AgMDmzUdgbQPWVlZKCoqQu/evWFlZYWrV6/i5cuXOHLkCDZs2IDJkydLLdGsWrUKFhYW+Oyzz4TH+vXrJ5VYGtMmk0wNeXl5TJs2Dbdv30ZAQABu3LgBPT09/Pzzz3j58iWKi4uFG6cT8l9KSkq11hNSVVVFhw4dMHPmTBw9ehSKioq1/mhJiqenJ3JycrBr165ax3fs2CHxWETR7I5fPp8PgUDQavaS7tu3L/bt24ecnBwcOXIEgwcPxrhx46ClpSXt0IiM6tWrF3r16oWcnBzk5+fD1tYWXbt2BfDuLsfW1lakHSHF6dKlS/Dx8UFkZCQUFBRqnTt69KhEYxFVs5NMbGwsCgoKMGHCBHHGIzFaWloIDw9H7969hZuZE/I+OTk5+Pj4wMPDA2pqati+fTuAd1NawsPD4eXlBVNT0wYHfIpTeHg4XF1dERoaWueKjp6enjI5SbLZScbMzAzdu3dHYmKiOOORqICAAAwZMkTaYRAZpqSkBGNjY+jo6AinoCgrK2Px4sXQ0NBAcHCwRJJMZGQkVqxYgStXrtS7dUpycjLncTSHzIyTYYwJd+P7L0VFRU7mGH388ccSv90lrc/IkSPRsWNHABDuKda9e3dUVFQ0OLVFXEJDQ+Hs7IyLFy9CX1+/3nKRkZGcx9IcMpNkACAuLg6TJk2Cs7MzzM3NUV1djfv376Nbt27YsGGD2Nu7ePEibGxsYGBgIPa6Sdvx/jbGhYWF+Omnn2Bubo6ePXvW2hubCwEBAfj111/x119/Nbo5oLm5OXJzczmNpzlkJsnUTGbU0dGBmZkZZs6cCQD48ssvcfHiRU7adHJyatbukaT9+uijj3Do0CHO26mursbmzZtx9epV3Lp1S6Sf0y+++ILzuJpDZpJMjf8O3y4rK4OdnR0nbRkbG3NSLyEtkZ+fD3t7e2hra+PmzZtQVVUV6brff/+d48iaRybHyTx79gyJiYmIjo6Gj4+PtMMhRGKuXbsGCwsLzJkzB76+viInGFkmc3cy76usrKxzoW9C2prXr1/D1dUVaWlpOHfuHAYPHiztkMRGJu9k+vbtC2NjY1haWmLlypXC469evcLatWsRGBgoxegIEZ/q6mp4e3vD1NQUJiYmiIiIaFMJBpDRJPO+rl274s6dO3j79i26desGbW1t2qaEtAk3b97E6NGjERcXh6ioKKxevbpNLj0iU49LjDEUFxcjLy8PhYWFEAgEePr0Kc6dOycc8NS5c2cpR0lI8/H5fJw6dQp79uyBnp4e/P392/wQCplJMowx3LlzB4cPHwYARERECM+tWrVKWmERIhbPnj3DH3/8AT8/P4wZMwZ+fn4YNGiQtMOSCJlJMnJycrS2C2lTXr9+jeDgYJw4cQLJyclwdHREZGQk+vTpI+3QJEpmkgwhrR2fz0dsbCyioqJw/fp1PHr0CJMnT8aaNWtgY2Pzwazp9qJVJZnS0lJhp29ZWVmbGENAWic+n4/U1FTExsYiLi4OcXFxyMjIwCeffAILCwts2rQJ5ubmwo7cV69eISAgAK6urlKOXPJaVZJRU1PDunXrpB0GaQdKS0tRUFCAnJwc5OTkIDs7G1lZWUhPT0dGRgb4fD4GDBiA4cOHw9TUFE5OTjA0NGzw06G6loBtD1pVkiGES0lJScLJkGpqatDQ0ICuri569eoFXV1dWFtbY9myZdDX12/S7H0ej4fjx48jKioK1tbWMDEx4eotyCRKMoT8vyFDhnAyGXfevHnYtm0b+Hw+vv76awQFBUFHRwcAkJubCx6P16bn0bW9kT+ENBNXA+GePn0KXV1dyMvLw8jICNHR0QDerU2zcuVK2dxaVowoyRDCsS5duggXrC8rKxMOvjt+/DjGjx8vzdAkgpIMIQB69OjB2ZIiu3fvxt69exEXF4f+/ftj6NChiI6OxrBhw4QLk7dllGQIAaCjo4OFCxdyUvf48eOxfv16mJiYYPPmzeDxeLh69SrKy8uRkZGBZ8+egcfjcdK2LGhRxy/NIyKk6V6/fo38/HwEBAQgJSUFfD4f2dnZ0NDQkHZonJBj7fXDe0IkiDEGgUDwwYL4Pj4+KC4uhouLi9s4aQ0AAADVSURBVJQi4x4lGUKkKDc3FwKBoE3PZ6IkQwjhFHX8EkI4RUmGEMIpSjKEEE5RkiGEcIqSDCGEU5RkCCGcoiRDCOEUJRlCCKcoyRBCOEVJhhDCKUoyhBBOUZIhhHCKkgwhhFOUZAghnKIkQwjhFCUZQginKMkQQjhFSYYQwilKMoQQTlGSIYRwipIMIYRTlGQIIZyiJEMI4RQlGUIIpyjJEEI4RUmGEMIpSjKEEE5RkiGEcIqSDCGEU5RkCCGcoiRDCOEUJRlCCKcoyRBCOEVJhhDCqf8DOpdgxM1lSDkAAAAASUVORK5CYII=)
The variation of gravitational field E and potential
as a function of x from the center of a uniform spherical mass distribution of radius R is correctly represented, from the following plots, by
![](data:image/png;base64,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)
physics-General
physics-
A (nonrotating) star collapses onto itself from an initial radius Ri with its mass remaining unchanged. Which curve in figure best gives the gravitational acceleration ag on the surface of the star as a function of the radius of the star during the collapse?
![](data:image/png;base64,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)
A (nonrotating) star collapses onto itself from an initial radius Ri with its mass remaining unchanged. Which curve in figure best gives the gravitational acceleration ag on the surface of the star as a function of the radius of the star during the collapse?
![](data:image/png;base64,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)
physics-General
physics-
Inside a charged thin conducting spherical shell of radius R, a point charge +Q is placed as shown in figure. The potential of the shell would be
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)
Inside a charged thin conducting spherical shell of radius R, a point charge +Q is placed as shown in figure. The potential of the shell would be
![](data:image/png;base64,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)
physics-General
physics-
A spherical hole of radius R2 is excavated from the asteroid of mass M as shown in the figure. The gravitational acceleration at a point on the surface of the asteroid just above the excavation is
![](data:image/png;base64,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)
A spherical hole of radius R2 is excavated from the asteroid of mass M as shown in the figure. The gravitational acceleration at a point on the surface of the asteroid just above the excavation is
![](data:image/png;base64,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)
physics-General
physics-
Net charge on the outer surface of conducting sphere is
Net charge on the outer surface of conducting sphere is
physics-General
maths-
where
are the co-factors of the elements
for
are the direction cosines of of three mutually perpendicular lines then
and
are
where
are the co-factors of the elements
for
are the direction cosines of of three mutually perpendicular lines then
and
are
maths-General
maths-
The point collinear with A(0,1,-2),B(3,l,-1) and C(m-3,-4) is
The point collinear with A(0,1,-2),B(3,l,-1) and C(m-3,-4) is
maths-General
maths-
The shortest distance between any two opposite edges of the tetrahedron formed by planes x+y=0,y+z=0,z+x=0,x+y+z=a is constant, equal to ___
The shortest distance between any two opposite edges of the tetrahedron formed by planes x+y=0,y+z=0,z+x=0,x+y+z=a is constant, equal to ___
maths-General
maths-
In a three dimensional coordinate system P,Q and R are images of a point
in the x-y the y-z and z-x planes, respectly. If G is the centroid of triangle P Q R then area of triangle AOG is (O is origin)
In a three dimensional coordinate system P,Q and R are images of a point
in the x-y the y-z and z-x planes, respectly. If G is the centroid of triangle P Q R then area of triangle AOG is (O is origin)
maths-General
maths-
The angle between the planes 2x-y+z=6,x+y+2z=7 is
The angle between the planes 2x-y+z=6,x+y+2z=7 is
maths-General
maths-
‘L’ is a line in the space given by the Equation
and
is a plane in the space given by the Equation x + y + z=1 then
‘L’ is a line in the space given by the Equation
and
is a plane in the space given by the Equation x + y + z=1 then
maths-General
maths-
The intercepts made by a plane, P , on the co-ordinate axes are a, b, c. The co-ordinate axes are rotated about the origin by an angle q. The plane P now makes the intercepts a¢ , b¢ , c¢ on the (new) co-ordinate axes. Which of the following statements is true?
The intercepts made by a plane, P , on the co-ordinate axes are a, b, c. The co-ordinate axes are rotated about the origin by an angle q. The plane P now makes the intercepts a¢ , b¢ , c¢ on the (new) co-ordinate axes. Which of the following statements is true?
maths-General
maths-
A ray of light is sent through the point P (1, 2, 3) and is reflected on the XY plane. If the reflected ray passes through the point Q (3, 2, 5) then the equation of the reflected ray is
A ray of light is sent through the point P (1, 2, 3) and is reflected on the XY plane. If the reflected ray passes through the point Q (3, 2, 5) then the equation of the reflected ray is
maths-General
maths-
The length of the rectangular field in 100m If its perimeter is 300m then what is its breadth?
The length of the rectangular field in 100m If its perimeter is 300m then what is its breadth?
maths-General
maths-
The side of a square field is 65m What is the length of the fence required all around it ?
The side of a square field is 65m What is the length of the fence required all around it ?
maths-General