Answer to Question #232190 in Physics for Soriano

Question #232190
A sphere carrying 5.5 NC of charge is placed 100mm from a small sphere carrying 4.0 NC of charge. Assume the spheres are small relative too the 100mm seperation.

At what position from the particles will the electric magnetic field magnitude be zero
1
Expert's answer
2021-09-02T10:41:36-0400

Let q1=5.5×109C, q2=4×109C, r=100mm=0.1mq_1 = 5.5\times 10^{-9}C, \space q_2 = 4\times 10^{-9}C, \space r = 100mm = 0.1m. The electric field produced by the first sphere at some distance xx from it is given as follows:


E1=kq1x2E_1 = k\dfrac{q_1}{x^2}

where kk is the constant and xx is the distance from the center of the first sphere. The second sphere produces the following field at the same point xx:


E2=kq2(rx)2E_2 = k\dfrac{q_2}{(r-x)^2}

These fields are directed in opposite direction, thus, the resulting field is zero if


E1=E2E_1 = E_2

Solving for xx, obtain:


kq1x2=kq2(rx)2q1(rx)2=q2x2q1r22q1rx+q1x2=q2x2(q1q2)x22q1rx+q1r2=01.5×109x21.1×109x+0.055×109=01.5x21.1x+0.055=0x10.054m=5.4cmx20.68m=68cmk\dfrac{q_1}{x^2} = k\dfrac{q_2}{(r-x)^2}\\ q_1(r - x)^2 = q_2x^2\\ q_1r^2 - 2q_1rx + q_1x^2 = q_2x^2\\ (q_1 - q_2)x^2 - 2q_1rx + q_1r^2 = 0\\ 1.5\times 10^{-9}x^2 - 1.1\times 10^{-9}x + 0.055\times 10^{-9} = 0\\ 1.5x^2 - 1.1x + 0.055 = 0\\ x_1 \approx 0.054m = 5.4cm\\ x_2 \approx 0.68m = 68cm

Answer. The electric magnetic field magnitude is zero at 5.4cm and 68 cm from the first particle.


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