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<meta property="og:title" content="There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge Q which has mass M. At the initial istant, the sphere is in rest. Certain particle of mass m and charge q (same sign of Q) is moved from infinite with speed v over line that join the holes. Determine the time which particle will be inside the sphere.">
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<meta property="og:description" content="There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge Q which has mass M. At the initial istant, the sphere is in rest. Certain particle of mass m and charge q (same sign of Q) is moved from infinite with speed v over line that join the holes. Determine the time which particle will be inside the sphere.">
<title>There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge Q which has mass M. At the initial istant, the sphere is in rest. Certain particle of mass m and charge q (same sign of Q) is moved from infinite with speed v over line that join the holes. Determine the time which particle will be inside the sphere.</title>
There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge $Q$ which has mass $M$. At the initial istant, the sphere is in rest. Certain particle of mass $m$ and charge $q$ (same sign of $Q$) is moved from infinite with speed $v$ over line that join the holes. Determine the time which particle will be inside the sphere.
</p>
<h3>Solution</h3>
<p>
Firstly, we consider the situation before $q$-particle penetrates $Q$-sphere. Since both bodies have equal-sign charges, $q$-particle will move with a deacceleration while $Q$-sphere will accelerate.
Applying Energy Conservation Law (from the beginning until $q$-particle reaches $Q$-sphere surface)
Inside of $Q$-sphere, there is no electric field, then both bodies have constant velocities while $q$-particle be inside $Q$-sphere (see Figure). The displacement of $q$-particle is
</p>
<center>
<figure>
<img src="7.1.17.png"
loading="lazy" alt="7.1.17" width="250" />
<figcaption>
Particle motion inside the sphere
</figcaption>
</figure>
</center>
<p>
$$\Delta S_q = \Delta S_Q+2R = v_q~T \;(3)$$
And the sphere's displacement is
$$\Delta S_Q = v_Q~T \;(4)$$
From $(4)$ in $(3)$, and separating $T$
$$T = \frac{2R}{v_q-v_Q} \;(5)$$
Separating $v_q$ from $(2)$
$$v_q = v - \frac{M}{m}v_Q \;(6)$$
Putting $(6)$ into $(1)$ and developing algebraically
$(7)$ is a second-degree equation or quadratic equation in $v_Q$ ($AX^2+BX+C=0$). We can solve this by Discriminant method or General-Solution formula for quadratic equations. The Discriminant is
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master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
<meta property="og:title" content="There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge Q which has mass M. At the initial istant, the sphere is in rest. Certain particle of mass m and charge q (same sign of Q) is moved from infinite with speed v over line that join the holes. Determine the time which particle will be inside the sphere.">
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<meta property="og:description" content="There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge Q which has mass M. At the initial istant, the sphere is in rest. Certain particle of mass m and charge q (same sign of Q) is moved from infinite with speed v over line that join the holes. Determine the time which particle will be inside the sphere.">
<title>There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge Q which has mass M. At the initial istant, the sphere is in rest. Certain particle of mass m and charge q (same sign of Q) is moved from infinite with speed v over line that join the holes. Determine the time which particle will be inside the sphere.</title>
There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge $Q$ which has mass $M$. At the initial istant, the sphere is in rest. Certain particle of mass $m$ and charge $q$ (same sign of $Q$) is moved from infinite with speed $v$ over line that join the holes. Determine the time which particle will be inside the sphere.
There are two diametrically opposed holes in a non-conductive thin-walled sphere with an uniformly distributed charge $Q$ which has mass $M$. At the initial istant, the sphere is in rest. Certain particle of mass $m$ and charge $q$ (same sign of $Q$) is moved from infinite with speed $v$ over line that join the holes. Determine the time which particle will be inside the sphere.
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
Firstly, we consider the situation before $q$-particle penetrates $Q$-sphere. Since both bodies have equal-sign charges, $q$-particle will move with a deacceleration while $Q$-sphere will accelerate.
Firstly, we consider the situation before $q$-particle penetrates $Q$-sphere. Since both bodies have equal-sign charges, $q$-particle will move with a deacceleration while $Q$-sphere will accelerate.
Applying Energy Conservation Law (from the beginning until $q$-particle reaches $Q$-sphere surface)
Applying Energy Conservation Law (from the beginning until $q$-particle reaches $Q$-sphere surface)
Inside of $Q$-sphere, there is no electric field, then both bodies have constant velocities while $q$-particle be inside $Q$-sphere (see Figure). The displacement of $q$-particle is
Inside of $Q$-sphere, there is no electric field, then both bodies have constant velocities while $q$-particle be inside $Q$-sphere (see Figure). The displacement of $q$-particle is
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="7.1.17.png"
<img src="7.1.17.png"
loading="lazy" alt="7.1.17" width="250" />
loading="lazy" alt="7.1.17" width="250" />
<figcaption>
<figcaption>
Particle motion inside the sphere
Particle motion inside the sphere
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
$$\Delta S_q = \Delta S_Q+2R = v_q~T \;(3)$$
$$\Delta S_q = \Delta S_Q+2R = v_q~T \;(3)$$
And the sphere's displacement is
And the sphere's displacement is
$$\Delta S_Q = v_Q~T \;(4)$$
$$\Delta S_Q = v_Q~T \;(4)$$
From $(4)$ in $(3)$, and separating $T$
From $(4)$ in $(3)$, and separating $T$
$$T = \frac{2R}{v_q-v_Q} \;(5)$$
$$T = \frac{2R}{v_q-v_Q} \;(5)$$
Separating $v_q$ from $(2)$
Separating $v_q$ from $(2)$
$$v_q = v - \frac{M}{m}v_Q \;(6)$$
$$v_q = v - \frac{M}{m}v_Q \;(6)$$
Putting $(6)$ into $(1)$ and developing algebraically
Putting $(6)$ into $(1)$ and developing algebraically
$(7)$ is a second-degree equation or quadratic equation in $v_Q$ ($AX^2+BX+C=0$). We can solve this by Discriminant method or General-Solution formula for quadratic equations. The Discriminant is
$(7)$ is a second-degree equation or quadratic equation in $v_Q$ ($AX^2+BX+C=0$). We can solve this by Discriminant method or General-Solution formula for quadratic equations. The Discriminant is
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
