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<h2 class="hd hd-2 unit-title">W1PS1: Unit vector decomposition for Coulomb's Law. </h2>
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Unit Vector Decomposition for Coulomb&#39;s law
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Two point charges are fixed on a grid as shown. The squares forming the grid have sides of length [mathjaxinline]a[/mathjaxinline]. Consider a coordinate system with unit vectors [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline] in the directions shown and with its origin at charge 1 ([mathjaxinline]q_1[/mathjaxinline]). </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]\hat{r}_{1,2}[/mathjaxinline], the unit vector pointing from charge 1 to charge 2. Express your answer in terms of hati for [mathjaxinline]\hat{i}[/mathjaxinline], hatj for [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline] as needed. <p style="display:inline">[mathjaxinline]\hat{r}_{1,2}=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w1_14_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part b)</b> If [mathjaxinline]q_1 = q[/mathjaxinline] and [mathjaxinline]q_2 = -2q[/mathjaxinline], where [mathjaxinline]q &gt; 0[/mathjaxinline], calculate the x and y components of the force exerted by charge [mathjaxinline]q_1[/mathjaxinline] on charge [mathjaxinline]q_2[/mathjaxinline] (where the x and y axes are in the [mathjaxinline]\hat{i}[/mathjaxinline] and [mathjaxinline]\hat{j}[/mathjaxinline] directions, respectively) in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]q[/mathjaxinline]. </p>
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<p style="display:inline">[mathjaxinline]F_ x=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]F_ y=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">W1PS2: Force between two charges.</h2>
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Force between two charges given their position, part 1.
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Two small objects are in the [mathjaxinline](x,y)[/mathjaxinline] plane as shown. Measured in meters from the zero of the coordinate system, the object of charge [mathjaxinline]q_1=+2q[/mathjaxinline] is at position [mathjaxinline]\vec{r}_1 = \left(1\; \hat{i} + 2\; \hat{j}\right)[/mathjaxinline] (m), and the object of charge [mathjaxinline]q_2=+5q[/mathjaxinline] is at position [mathjaxinline]\vec{r}_2=\left(4\; \hat{i} + 3\; \hat{j}\right)[/mathjaxinline] (m). If [mathjaxinline]q= 2 \; \mu \mbox{C}[/mathjaxinline] and the Coulomb constant is [mathjaxinline]k = 9\times 10^9 \mbox{N}\cdot \mbox{C}^{-2}\cdot \mbox{m}^2[/mathjaxinline], calculate the x component of the force exerted by charge 1 on charge 2. </p>
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<p style="display:inline">[mathjaxinline]F_ x=[/mathjaxinline]</p>
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Force between two charges given their position, part 2.
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Calculate the y component of the force exerted by charge 1 on charge 2. <p style="display:inline">[mathjaxinline]F_ y=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w1_9_b_2_1" class=" capa_inputtype inline textline">
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<h2 class="hd hd-2 unit-title">W1PS3: Superposition</h2>
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Force on a third charge, Part 1.
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Two point charges [mathjaxinline]q_1=-2q[/mathjaxinline] and [mathjaxinline]q_2=+q[/mathjaxinline], where [mathjaxinline]q&gt;0[/mathjaxinline], are separated by a distance [mathjaxinline]a[/mathjaxinline] in the [mathjaxinline]\hat{j}[/mathjaxinline] direction. A third charge, [mathjaxinline]q_3=+5q[/mathjaxinline], is placed a distance [mathjaxinline]2a[/mathjaxinline] away from [mathjaxinline]q_2[/mathjaxinline] in the [mathjaxinline]\hat{i}[/mathjaxinline] direction as shown. </p>
<p><b class="bfseries">(Part a)</b> Calculate, [mathjaxinline]\vec{F}_{13}[/mathjaxinline], the force exerted by the charge [mathjaxinline]q_1[/mathjaxinline] on the charge [mathjaxinline]q_3[/mathjaxinline]. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]q[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], hati for [mathjaxinline]\hat{i}[/mathjaxinline] and hatj for [mathjaxinline]\hat{j}[/mathjaxinline] as needed. </p>
<p>
<p style="display:inline">[mathjaxinline]\vec{F}_{13}=[/mathjaxinline]</p>
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Force on a third charge, Part 2
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<p><b class="bfseries">(Part b)</b> Calculate, [mathjaxinline]\vec{F}_{23}[/mathjaxinline], the force exerted by charge 2 on charge 3. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]q[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], hati for [mathjaxinline]\hat{i}[/mathjaxinline] and hatj for [mathjaxinline]\hat{j}[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\vec{F}_{23}=[/mathjaxinline]</p>
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Force on a third charge, Part 3
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<p><b class="bfseries">(Part c)</b> Calculate, [mathjaxinline]\vec{F}_{3}[/mathjaxinline], the total force exerted by both charges 1 and 2 on charge 3. Express your answer in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]q[/mathjaxinline], [mathjaxinline]a[/mathjaxinline], hati for [mathjaxinline]\hat{i}[/mathjaxinline] and hatj for [mathjaxinline]\hat{j}[/mathjaxinline] as needed. </p>
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<p style="display:inline">[mathjaxinline]\vec{F}_{3}=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">W1PS4: Unit vector decomposition for Electric Field. </h2>
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Unit Vector Decomposition for Electric Field
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<p>
Two point charges are fixed on a grid as shown. The squares forming the grid have sides of length [mathjaxinline]a[/mathjaxinline]. Consider a coordinate system with unit vectors [mathjaxinline]\hat{i}[/mathjaxinline], [mathjaxinline]\hat{j}[/mathjaxinline] in the directions shown. </p>
<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]\hat{r}_{1,P}[/mathjaxinline], the unit vector pointing from charge 1 to the point [mathjaxinline]P[/mathjaxinline]. Express your answer in terms of hati for [mathjaxinline]\hat{i}[/mathjaxinline], hatj for [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline] as needed. <p style="display:inline">[mathjaxinline]\hat{r}_{1,P}=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_checkpoint_w1_14Mod_2_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part a)</b> Calculate [mathjaxinline]\hat{r}_{2,P}[/mathjaxinline], the unit vector pointing from charge 2 to the point [mathjaxinline]P[/mathjaxinline]. Express your answer in terms of hati for [mathjaxinline]\hat{i}[/mathjaxinline], hatj for [mathjaxinline]\hat{j}[/mathjaxinline], and [mathjaxinline]a[/mathjaxinline] as needed. <p style="display:inline">[mathjaxinline]\hat{r}_{2,P}=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_checkpoint_w1_14Mod_3_1" class="text-input-dynamath capa_inputtype inline textline">
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<p><b class="bfseries">(Part b)</b> If [mathjaxinline]q_1 = q[/mathjaxinline] and [mathjaxinline]q_2 = -2q[/mathjaxinline], where [mathjaxinline]q &gt; 0[/mathjaxinline], calculate the x and y components of the total electric field (where the x and y axes are in the [mathjaxinline]\hat{i}[/mathjaxinline] and [mathjaxinline]\hat{j}[/mathjaxinline] directions, respectively) created by charges [mathjaxinline]q_1[/mathjaxinline] and [mathjaxinline]q_2[/mathjaxinline] at the point [mathjaxinline]P[/mathjaxinline] in terms of [mathjaxinline]k[/mathjaxinline], [mathjaxinline]a[/mathjaxinline] and [mathjaxinline]q[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]E_ x=[/mathjaxinline]</p>
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<p>
<p style="display:inline">[mathjaxinline]E_ y=[/mathjaxinline]</p>
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